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[Axiom-developer] 20080427.01.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080427.01.tpd.patch (CATS Schaums-Axiom equivalence testing (29-34)) |
|
Date: |
Mon, 28 Apr 2008 01:53:34 -0500 |
14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
14:671 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
In these two problems Axiom constructs a closed form for the integral
whereas Schaums shows a series expansion. Axiom makes the claim that
if there exists a closed form for an integral in elementary terms then
Axiom will return that form. If it returns the original integral then
no closed form exists. Thus, Axiom is a decision procedure. In these
two cases, Axiom has constructed a closed form.
Also, in
14:661 Schaums and Axiom agree
I believe that my copy of Schaums has a typo and that instead of
integral acoth(x/a) = x*acoth(x)+a/2*log(x^2-a^2)
the correct result should be
integral acoth(x/a) = x*acoth(x/a)+a/2*log(x^2-a^2)
This has been fixed proactively but needs to be verified against a
later copy of Schaums. It appears that I've gotten so old that the
equations have changed since my college mathematics education :-)
At this point I have checked Axiom against all of the indefinite
integrals in Schaums. Frankly, I'm quite impressed with the Axiom
team (Trager, Bronstein, et.al.) that created this code (Note that
I did not author any of the integration code). By any measure this
is an impressive showing.
Of the forms that Axiom could not simplify I have two further approaches.
The first approach will be to check the answers using Maxima/MMA/Maple.
Note that I believe the problems are mostly simplification issues.
The second approach is to spend some time thinking about a better,
more general method for automating the simplification, including using
proviso forms to capture assumptions. I have a graph-based procedure
that I'll explain and codify in a different pamphlet.
Another point is that there were a couple simplifications that ended
up with x only in the imaginary portion of the answer. This indicates
to me that there is a sign assumption in the code (see 14.666, around
-S 124 where I expand this more fully in a clearer example). These
assumptions need to be pushed forward in provisos.
Much remains to be done with CATS but this exercise has left me with
the impression that Axiom's integration routines are rock solid and
can be relied on with confidence. The 4Ms could benefit from doing
the same exercise but I don't have the time available to do that yet.
Ideally it could be shown that all of the CASs are "plug-compatible",
giving known-good answers to published problems.
schaum29.input.pamphlet
14:590 Schaums and Axiom agree
14:591 Axiom cannot simplify this expression
14:592 Axiom cannot simplify this expression
14:593 Axiom cannot simplify this expression
14:594 Schaums and Axiom agree
14:595 Schaums and Axiom agree
14:596 Axiom cannot simplify this expression
14:597 Schaums and Axiom agree
14:598 Axiom cannot simplify this expression
14:599 Axiom cannot simplify this expression
14:600 Axiom cannot simplify this expression
14:601 Schaums and Axiom agree
14:602 Schaums and Axiom agree
14:603 Schaums and Axiom agree
schaum30.input.pamphlet
14:604 Schaums and Axiom differ by a constant
14:605 Schaums and Axiom differ by a constant
14:606 Axiom cannot simplify this expression
14:607 Axiom cannot simplify this expression
14:608 Schaums and Axiom agree
14:609 Schaums and Axiom differ by a constant
14:610 Axiom cannot compute this integral
14:611 Schaums and Axiom differ by a constant
14:612 Axiom cannot compute this integral
14:613 Schaums and Axiom differ by a constant
14:614 Axiom cannot compute this integral
schaum31.input.pamphlet
14:615 Schaums and Axiom differ by a constant
14:616 Schaums and Axiom differ by a constant
14:617 Axiom cannot simplify this expression
14:618 Schaums and Axiom agree
14:619 Schaums and Axiom agree
14:620 Schaums and Axiom differ by a constant
14:621 Axiom cannot compute this integral
14:622 Schaums and Axiom differ by a constant
14:623 Axiom cannot compute this integral
14:624 Schaums and Axiom differ by a constant
14:625 Axiom cannot compute this integral
schaum32.input.pamphlet
14:626 Schaums and Axiom agree
14:627 Schaums and Axiom differ by a constant
14:628 Axiom cannot simplify this expression
14:629 Schaums and Axiom agree
14:630 Schaums and Axiom agree
14:631 Axiom cannot compute this integral
14:632 Schaums and Axiom differ by a constant
14:633 Axiom cannot compute this integral
14:634 Schaums and Axiom agree
14:635 Axiom cannot compute this integral
schaum33.input.pamphlet
14:636 Schaums and Axiom agree
14:637 Axiom cannot simplify this expression
14:638 Axiom cannot simplify this expression
14:639 Schaums and Axiom agree
14:640 Schaums and Axiom agree
14:641 Axiom cannot compute this integral
14:642 Axiom cannot simplify this expression
14:643 Axiom cannot compute this integral
14:644 Schaums and Axiom differ by a constant
14:645 Axiom cannot compute this integral
schaum34.input.pamphlet
14:646 Schaums and Axiom agree
14:647 Schaums and Axiom agree
14:648 Schaums and Axiom agree
14:649 Axiom cannot compute this integral
14:650 Schaums and Axiom differ by a constant
14:651 Schaums and Axiom agree
14:652 Schaums and Axiom agree
14:653 Schaums and Axiom agree
14:654 Axiom cannot compute this integral
14:655 Axiom cannot simplify these expressions
14:656 Schaums and Axiom differ by a constant
14:657 Schaums and Axiom agree
14:658 Schaums and Axiom differ by a constant
14:659 Axiom cannot compute this integral
14:660 Schaums and Axiom agree
14:661 Schaums and Axiom agree
14:662 Schaums and Axiom agree
14:663 Schaums and Axiom agree
14:664 Axiom cannot compute this integral
14:665 Schaums and Axiom agree
14:666 Schaums and Axiom agree
14:667 Schaums and Axiom differ by a constant
14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
14:669 Axiom cannot simplify these expressions
14:670 Axiom cannot simplify these expressions
14:671 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
14:672 Axiom cannot compute this integral
14:673 Axiom cannot compute this integral
14:674 Axiom cannot compute this integral
14:675 Axiom cannot compute this integral
14:676 Axiom cannot compute this integral
14:677 Axiom cannot compute this integral
===========================================================================
diff --git a/src/input/schaum28.input.pamphlet
b/src/input/schaum28.input.pamphlet
index 5c68b06..3ba0bfc 100644
--- a/src/input/schaum28.input.pamphlet
+++ b/src/input/schaum28.input.pamphlet
@@ -234,6 +234,10 @@ aa:=integrate(x/cosh(a*x),x)
\section{\cite{1}:14.569~~~~~$\displaystyle
\int{\cosh^2{ax}}~dx$}
$$\int{\cosh^2{ax}}=
+\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2a}
+$$
+Note that the Schaums print edition (1968 printing 3) has a typo:
+$$\int{\cosh^2{ax}}=
\frac{x}{2}+\frac{\sinh{ax}\cosh{ax}}{2}
$$
<<*>>=
@@ -250,40 +254,21 @@ aa:=integrate(cosh(a*x)^2,x)
--E
--S 18
-bb:=x/2+(sinh(a*x)*cosh(a*x))/2
+bb:=x/2+(sinh(a*x)*cosh(a*x))/(2*a)
--R
---R cosh(a x)sinh(a x) + x
---R (2) ----------------------
---R 2
+--R cosh(a x)sinh(a x) + a x
+--R (2) ------------------------
+--R 2a
--R Type: Expression
Integer
--E
---S 19
+--S 19 14:569 Schaums and Axiom agree
cc:=aa-bb
--R
---R (- a + 1)cosh(a x)sinh(a x)
---R (3) ---------------------------
---R 2a
+--R (3) 0
--R Type: Expression
Integer
--E
---S 20
-sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
---R
---I %B sinh(y + x) - %B sinh(y - x)
---I (4) %B cosh(y)sinh(x) == -------------------------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 21 14:569 SCHAUMS AND AXIOM DISAGREE?
-dd:=sinhcoshrule cc
---R
---R (- a + 1)sinh(2a x)
---R (5) -------------------
---R 4a
---R Type: Expression
Integer
---E
@
\section{\cite{1}:14.570~~~~~$\displaystyle
@@ -294,7 +279,7 @@ $$
<<*>>=
)clear all
---S 22
+--S 20
aa:=integrate(x*cosh(a*x)^2,x)
--R
--R
@@ -306,7 +291,7 @@ aa:=integrate(x*cosh(a*x)^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 23
+--S 21
bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)
--R
--R 2 2
@@ -317,7 +302,7 @@ bb:=x^2/4+(x*sinh(2*a*x))/(4*a)-cosh(2*a*x)/(8*a^2)
--R Type: Expression
Integer
--E
---S 24
+--S 22
cc:=aa-bb
--R
--R (3)
@@ -332,7 +317,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 25
+--S 23
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -341,7 +326,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 26
+--S 24
dd:=sinhsqrrule cc
--R
--R (5)
@@ -353,7 +338,7 @@ dd:=sinhsqrrule cc
--R Type: Expression
Integer
--E
---S 27
+--S 25
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -362,7 +347,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 28
+--S 26
ee:=coshsqrrule dd
--R
--R - x sinh(2a x) + 2x cosh(a x)sinh(a x)
@@ -371,7 +356,7 @@ ee:=coshsqrrule dd
--R Type: Expression
Integer
--E
---S 29
+--S 27
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %S sinh(y + x) - %S sinh(y - x)
@@ -380,7 +365,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) ==
1/2*(sinh(x+y)+sinh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 30 14:570 Schaums and Axiom agree
+--S 28 14:570 Schaums and Axiom agree
ff:=sinhcoshrule ee
--R
--R (9) 0
@@ -396,7 +381,7 @@ $$
<<*>>=
)clear all
---S 31
+--S 29
aa:=integrate(1/cosh(a*x)^2,x)
--R
--R
@@ -407,7 +392,7 @@ aa:=integrate(1/cosh(a*x)^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 32
+--S 30
bb:=tanh(a*x)/a
--R
--R tanh(a x)
@@ -416,7 +401,7 @@ bb:=tanh(a*x)/a
--R Type: Expression
Integer
--E
---S 33
+--S 31
cc:=aa-bb
--R
--R 2 2
@@ -425,155 +410,12 @@ cc:=aa-bb
--R 2 2
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Expression
Integer
---E
-
---S 34
-tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
---R
---R sinh(x)
---R (4) tanh(x) == -------
---R cosh(x)
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 35
-dd:=tanhrule cc
---R
---R (5)
---R 3 2 2
---R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x)
---R +
---R - 2cosh(a x)
---R /
---R 2 2 3
---R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a
cosh(a x)
---R Type: Expression
Integer
---E
-
---S 36
-sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
---R
---R 2 cosh(2x) - 1
---R (6) sinh(x) == ------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 37
-ee:=sinhsqrrule dd
---R
---R (7)
---R 3 2
---R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x)
---R +
---R - 2cosh(a x)
---R /
---R 2 3
---R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
---R +
---R a cosh(a x)
---R Type: Expression
Integer
---E
-
---S 38
-coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
---R
---R 2 cosh(2x) + 1
---R (8) cosh(x) == ------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 39
-ff:=coshsqrrule ee
---R
---R (9)
---R 3
---R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x)
---R +
---R - 2cosh(a x)
---R /
---R 3
---R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
---R +
---R a cosh(a x)
---R Type: Expression
Integer
---E
-
---S 40
-sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x))
---R
---R 3 sinh(3x) - 3sinh(x)
---R (10) sinh(x) == -------------------
---R 4
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 41
-gg:=sinhcuberule ff
---R
---R (11)
---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
---R +
---R - 4cosh(a x)
---R /
---R
3
---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x)
---R +
---R 2a cosh(a x)
---R Type: Expression
Integer
---E
-
---S 42
-coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
---R
---R 3 cosh(3x) - 3cosh(x)
---R (12) cosh(x) == -------------------
---R 4
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 43
-hh:=coshcuberule gg
---R
---R (13)
---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
---R +
---R - 4cosh(a x)
---R /
---R (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x)
---R +
---R - a cosh(a x)
---R Type: Expression
Integer
---E
-
---S 44
-coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
---R
---I %U cosh(y + x) + %U cosh(y - x)
---I (14) %U cosh(x)cosh(y) == -------------------------------
---R 2
---R Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 45
-ii:=coshcoshrule hh
---R
---R (15)
---R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x)
---R ----------------------------------------------------------------------
---R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x)
---R Type: Expression
Integer
---E
-
---S 46 14:571 SCHAUMS AND AXIOM DISAGREE?
-jj:=complexNormalize ii
+--S 32
+dd:=complexNormalize cc
--R
---R a x 2
---R - 2(%e ) - 2
---R (16) ---------------
---R a x 2
---R 2a (%e ) - a
+--R 1
+--R (4) - -
+--R a
--R Type: Expression
Integer
--E
@
@@ -586,7 +428,7 @@ $$
<<*>>=
)clear all
---S 47
+--S 33
aa:=integrate(cosh(a*x)*cosh(p*x),x)
--R
--R
@@ -597,7 +439,7 @@ aa:=integrate(cosh(a*x)*cosh(p*x),x)
--R Type: Union(Expression
Integer,...)
--E
---S 48
+--S 34
bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p))
--R
--R (p - a)x sinh(p + a) + (p + a)x sinh(p - a)
@@ -607,7 +449,7 @@ bb:=(sinh(a-p)*x)/(2*(a-p))+(sinh(a+p)*x)/(2*(a+p))
--R Type: Expression
Integer
--E
---S 49
+--S 35
cc:=aa-bb
--R
--R (3)
@@ -627,7 +469,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 50
+--S 36
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -636,7 +478,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 51
+--S 37
dd:=sinhsqrrule cc
--R
--R (5)
@@ -653,7 +495,7 @@ dd:=sinhsqrrule cc
--R Type: Expression
Integer
--E
---S 52
+--S 38
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -662,7 +504,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 53
+--S 39
ee:=coshsqrrule dd
--R
--R (7)
@@ -675,7 +517,7 @@ ee:=coshsqrrule dd
--R Type: Expression
Integer
--E
---S 54
+--S 40
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %V sinh(y + x) - %V sinh(y - x)
@@ -684,7 +526,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) ==
1/2*(sinh(x+y)+sinh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 55 14:572 Schaums and Axiom agree
+--S 41 14:572 Schaums and Axiom agree
ff:=sinhcoshrule ee
--R
--R (9)
@@ -706,7 +548,7 @@ $$
<<*>>=
)clear all
---S 56
+--S 42
aa:=integrate(cosh(a*x)*sin(p*x),x)
--R
--R
@@ -724,7 +566,7 @@ aa:=integrate(cosh(a*x)*sin(p*x),x)
--R Type: Union(Expression
Integer,...)
--E
---S 57
+--S 43
bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2)
--R
--R a sin(p x)sinh(a x) - p cos(p x)cosh(a x)
@@ -734,7 +576,7 @@ bb:=(a*sinh(a*x)*sin(p*x)-p*cosh(a*x)*cos(p*x))/(a^2+p^2)
--R Type: Expression
Integer
--E
---S 58
+--S 44
cc:=aa-bb
--R
--R (3)
@@ -749,7 +591,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 59
+--S 45
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -758,7 +600,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 60
+--S 46
dd:=coshsqrrule cc
--R
--R (5)
@@ -772,7 +614,7 @@ dd:=coshsqrrule cc
--R Type: Expression
Integer
--E
---S 61
+--S 47
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -781,7 +623,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 62 14:573 Schaums and Axiom agree
+--S 48 14:573 Schaums and Axiom agree
ee:=sinhsqrrule dd
--R
--R (7) 0
@@ -797,7 +639,7 @@ $$
<<*>>=
)clear all
---S 63
+--S 49
aa:=integrate(cosh(a*x)*cos(p*x),x)
--R
--R
@@ -815,7 +657,7 @@ aa:=integrate(cosh(a*x)*cos(p*x),x)
--R Type: Union(Expression
Integer,...)
--E
---S 64
+--S 50
bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2)
--R
--R a cos(p x)sinh(a x) + p cosh(a x)sin(p x)
@@ -825,7 +667,7 @@ bb:=(a*sinh(a*x)*cos(p*x)+p*cosh(a*x)*sin(p*x))/(a^2+p^2)
--R Type: Expression
Integer
--E
---S 65
+--S 51
cc:=aa-bb
--R
--R (3)
@@ -840,7 +682,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 66
+--S 52
coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R
--R 2 cosh(2x) + 1
@@ -849,7 +691,7 @@ coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 67
+--S 53
dd:=coshsqrrule cc
--R
--R (5)
@@ -863,7 +705,7 @@ dd:=coshsqrrule cc
--R Type: Expression
Integer
--E
---S 68
+--S 54
sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R
--R 2 cosh(2x) - 1
@@ -872,7 +714,7 @@ sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 69 14:574 Schaums and Axiom agree
+--S 55 14:574 Schaums and Axiom agree
ee:=sinhsqrrule dd
--R
--R (7) 0
@@ -888,7 +730,7 @@ $$
<<*>>=
)clear all
---S 70
+--S 56
aa:=integrate(1/(cosh(a*x)+1),x)
--R
--R
@@ -898,7 +740,7 @@ aa:=integrate(1/(cosh(a*x)+1),x)
--R Type: Union(Expression
Integer,...)
--E
---S 71
+--S 57
bb:=1/a*tanh((a*x)/2)
--R
--R a x
@@ -909,7 +751,7 @@ bb:=1/a*tanh((a*x)/2)
--R Type: Expression
Integer
--E
---S 72
+--S 58
cc:=aa-bb
--R
--R a x
@@ -920,7 +762,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 73
+--S 59
tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
--R
--R sinh(x)
@@ -929,7 +771,7 @@ tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 74
+--S 60
dd:=tanhrule cc
--R
--R a x a x a x
@@ -942,7 +784,7 @@ dd:=tanhrule cc
--R Type: Expression
Integer
--E
---S 75
+--S 61
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %B sinh(y + x) - %B sinh(y - x)
@@ -951,7 +793,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) ==
1/2*(sinh(x+y)+sinh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 76
+--S 62
ee:=sinhcoshrule dd
--R
--R 3a x a x a x a x
@@ -964,7 +806,7 @@ ee:=sinhcoshrule dd
--R Type: Expression
Integer
--E
---S 77
+--S 63
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %B cosh(y + x) - %B cosh(y - x)
@@ -973,7 +815,7 @@
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 78
+--S 64
ff:=sinhsinhrule ee
--R
--R 3a x a x 3a x a x
@@ -986,7 +828,7 @@ ff:=sinhsinhrule ee
--R Type: Expression
Integer
--E
---S 79
+--S 65
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BC cosh(y + x) + %BC cosh(y - x)
@@ -995,7 +837,7 @@
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 80 14:575 Schaums and Axiom differ by a constant
+--S 66 14:575 Schaums and Axiom differ by a constant
gg:=coshcoshrule ff
--R
--R 1
@@ -1013,7 +855,7 @@ $$
<<*>>=
)clear all
---S 81
+--S 67
aa:=integrate(1/(cosh(a*x)-1),x)
--R
--R
@@ -1023,7 +865,7 @@ aa:=integrate(1/(cosh(a*x)-1),x)
--R Type: Union(Expression
Integer,...)
--E
---S 82
+--S 68
bb:=-1/a*coth((a*x)/2)
--R
--R a x
@@ -1034,7 +876,7 @@ bb:=-1/a*coth((a*x)/2)
--R Type: Expression
Integer
--E
---S 83
+--S 69
cc:=aa-bb
--R
--R a x a x
@@ -1045,7 +887,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 84
+--S 70
cothrule:=rule(coth(x) == cosh(x)/sinh(x))
--R
--R cosh(x)
@@ -1054,7 +896,7 @@ cothrule:=rule(coth(x) == cosh(x)/sinh(x))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 85
+--S 71
dd:=cothrule cc
--R
--R a x a x a x a x
@@ -1067,7 +909,7 @@ dd:=cothrule cc
--R Type: Expression
Integer
--E
---S 86
+--S 72
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %BD sinh(y + x) - %BD sinh(y - x)
@@ -1076,7 +918,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) ==
1/2*(sinh(x+y)+sinh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 87
+--S 73
ee:=sinhcoshrule dd
--R
--R 3a x a x a x a x
@@ -1089,7 +931,7 @@ ee:=sinhcoshrule dd
--R Type: Expression
Integer
--E
---S 88
+--S 74
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %BE cosh(y + x) - %BE cosh(y - x)
@@ -1098,7 +940,7 @@
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 89
+--S 75
ff:=sinhsinhrule ee
--R
--R 3a x a x a x a x
@@ -1111,7 +953,7 @@ ff:=sinhsinhrule ee
--R Type: Expression
Integer
--E
---S 90
+--S 76
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BF cosh(y + x) + %BF cosh(y - x)
@@ -1120,7 +962,7 @@
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 91 14:576 Schaums and Axiom differ by a constant
+--S 77 14:576 Schaums and Axiom differ by a constant
gg:=coshcoshrule ff
--R
--R 1
@@ -1138,7 +980,7 @@ $$
<<*>>=
)clear all
---S 92
+--S 78
aa:=integrate(x/(cosh(a*x)+1),x)
--R
--R
@@ -1152,7 +994,7 @@ aa:=integrate(x/(cosh(a*x)+1),x)
--R Type: Union(Expression
Integer,...)
--E
---S 93
+--S 79
bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2))
--R
--R a x a x
@@ -1164,7 +1006,7 @@ bb:=x/a*tanh((a*x)/2)-2/a^2*log(cosh((a*x)/2))
--R Type: Expression
Integer
--E
---S 94
+--S 80
cc:=aa-bb
--R
--R (3)
@@ -1185,7 +1027,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 95
+--S 81
tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
--R
--R sinh(x)
@@ -1194,7 +1036,7 @@ tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 96
+--S 82
dd:=tanhrule cc
--R
--R (5)
@@ -1222,7 +1064,7 @@ dd:=tanhrule cc
--R Type: Expression
Integer
--E
---S 97
+--S 83
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BG cosh(y + x) + %BG cosh(y - x)
@@ -1231,7 +1073,7 @@
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 98
+--S 84
ee:=coshcoshrule dd
--R
--R (7)
@@ -1259,7 +1101,7 @@ ee:=coshcoshrule dd
--R Type: Expression
Integer
--E
---S 99
+--S 85
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %BH sinh(y + x) - %BH sinh(y - x)
@@ -1268,7 +1110,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) ==
1/2*(sinh(x+y)+sinh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 100
+--S 86
ff:=sinhcoshrule ee
--R
--R (9)
@@ -1296,7 +1138,7 @@ ff:=sinhcoshrule ee
--R Type: Expression
Integer
--E
---S 101
+--S 87
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %BI cosh(y + x) - %BI cosh(y - x)
@@ -1305,7 +1147,7 @@
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 102
+--S 88
gg:=sinhsinhrule ff
--R
--R a x
@@ -1317,7 +1159,7 @@ gg:=sinhsinhrule ff
--R Type: Expression
Integer
--E
---S 103 14:577 Schaums and Axiom differ by a constant
+--S 89 14:577 Schaums and Axiom differ by a constant
complexNormalize gg
--R
--R 2log(2)
@@ -1336,7 +1178,7 @@ $$
<<*>>=
)clear all
---S 104
+--S 90
aa:=integrate(x/(cosh(a*x)-1),x)
--R
--R
@@ -1350,7 +1192,7 @@ aa:=integrate(x/(cosh(a*x)-1),x)
--R Type: Union(Expression
Integer,...)
--E
---S 105
+--S 91
bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2))
--R
--R a x a x
@@ -1362,7 +1204,7 @@ bb:=-x/a*coth((a*x)/2)+2/a^2*log(sinh((a*x)/2))
--R Type: Expression
Integer
--E
---S 106
+--S 92
cc:=aa-bb
--R
--R (3)
@@ -1383,7 +1225,7 @@ cc:=aa-bb
--R Type: Expression
Integer
--E
---S 107
+--S 93
cothrule:=rule(coth(x) == cosh(x)/sinh(x))
--R
--R cosh(x)
@@ -1392,7 +1234,7 @@ cothrule:=rule(coth(x) == cosh(x)/sinh(x))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 108
+--S 94
dd:=cothrule cc
--R
--R (5)
@@ -1420,7 +1262,7 @@ dd:=cothrule cc
--R Type: Expression
Integer
--E
---S 109
+--S 95
sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
--R
--I %BJ sinh(y + x) - %BJ sinh(y - x)
@@ -1429,7 +1271,7 @@ sinhcoshrule:=rule(sinh(x)*cosh(y) ==
1/2*(sinh(x+y)+sinh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 110
+--S 96
ee:=sinhcoshrule dd
--R
--R (7)
@@ -1457,7 +1299,7 @@ ee:=sinhcoshrule dd
--R Type: Expression
Integer
--E
---S 111
+--S 97
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R
--I %BK cosh(y + x) - %BK cosh(y - x)
@@ -1466,7 +1308,7 @@
sinhsinhrule:=rule(sinh(x)*sinh(y)==1/2*(cosh(x+y)-cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 112
+--S 98
ff:=sinhsinhrule ee
--R
--R (9)
@@ -1494,7 +1336,7 @@ ff:=sinhsinhrule ee
--R Type: Expression
Integer
--E
---S 113
+--S 99
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R
--I %BL cosh(y + x) + %BL cosh(y - x)
@@ -1503,7 +1345,7 @@
coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
--R Type: RewriteRule(Integer,Integer,Expression
Integer)
--E
---S 114
+--S 100
gg:=coshcoshrule ff
--R
--R a x
@@ -1515,7 +1357,7 @@ gg:=coshcoshrule ff
--R Type: Expression
Integer
--E
---S 115 14:578 Schaums and Axiom differ by a constant
+--S 101 14:578 Schaums and Axiom differ by a constant
hh:=complexNormalize gg
--R
--R 2log(2)
@@ -1534,7 +1376,7 @@ $$
<<*>>=
)clear all
---S 116
+--S 102
aa:=integrate(1/(cosh(a*x)+1)^2,x)
--R
--R
@@ -1552,7 +1394,7 @@ aa:=integrate(1/(cosh(a*x)+1)^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 117
+--S 103
bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3
--R
--R a x 3 a x
@@ -1563,7 +1405,7 @@ bb:=1/(2*a)*tanh((a*x)/2)-1/(6*a)*tanh((a*x)/2)^3
--R Type: Expression
Integer
--E
---S 118 14:579 Axiom cannot compute this integral
+--S 104 14:579 Axiom cannot compute this integral
cc:=aa-bb
--R
--R (3)
@@ -1614,7 +1456,7 @@ $$
<<*>>=
)clear all
---S 119
+--S 105
aa:=integrate(1/(cosh(a*x)-1)^2,x)
--R
--R
@@ -1632,7 +1474,7 @@ aa:=integrate(1/(cosh(a*x)-1)^2,x)
--R Type: Union(Expression
Integer,...)
--E
---S 120
+--S 106
bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3
--R
--R a x 3 a x
@@ -1643,7 +1485,7 @@ bb:=1/(2*a)*coth((a*x)/2)-1/(6*a)*coth((a*x)/2)^3
--R Type: Expression
Integer
--E
---S 121 14:580 Axiom cannot compute this integral
+--S 107 14:580 Axiom cannot compute this integral
cc:=aa-bb
--R
--R (3)
@@ -1702,7 +1544,7 @@ $$
<<*>>=
)clear all
---S 122
+--S 108
aa:=integrate(1/(p+q*cosh(a*x)),x)
--R
--R
@@ -1744,7 +1586,7 @@ aa:=integrate(1/(p+q*cosh(a*x)),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 123
+--S 109
bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2))
--R
--R a x
@@ -1760,7 +1602,7 @@
bb1:=2/(a*sqrt(q^2-p^2))*atan((q*%e^(a*x)+p)/sqrt(q^2-p^2))
--R Type: Expression
Integer
--E
---S 124
+--S 110
bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2-q^2)))
--R
--R +---------+
@@ -1777,7 +1619,7 @@
bb2:=1/(a*sqrt(p^2-q^2))*log((q*%e^(a*x)+p-sqrt(p^2-q^2))/(q*%e^(a*x)+p+sqrt(p^2
--R Type: Expression
Integer
--E
---S 125
+--S 111
cc1:=aa.1-bb1
--R
--R (4)
@@ -1817,7 +1659,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 126
+--S 112
cc2:=aa.2-bb1
--R
--R +-------+
@@ -1834,7 +1676,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 127
+--S 113
cc3:=aa.1-bb2
--R
--R (6)
@@ -1871,7 +1713,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 128 14:581 Axiom cannot simplify this expression
+--S 114 14:581 Axiom cannot simplify this expression
cc4:=aa.2-bb2
--R
--R (7)
@@ -1907,7 +1749,7 @@ $$
<<*>>=
)clear all
---S 129
+--S 115
aa:=integrate(1/(p+q*cosh(a*x))^2,x)
--R
--R
@@ -1989,7 +1831,7 @@ aa:=integrate(1/(p+q*cosh(a*x))^2,x)
--R Type: Union(List Expression
Integer,...)
--E
---S 130
+--S 116
t1:=integrate(1/(p+q*cosh(a*x)),x)
--R
--R (2)
@@ -2030,7 +1872,7 @@ t1:=integrate(1/(p+q*cosh(a*x)),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 131
+--S 117
bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1
--R
--R (3)
@@ -2066,7 +1908,7 @@
bb1:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.1
--R Type: Expression
Integer
--E
---S 132
+--S 118
bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2
--R
--R (4)
@@ -2087,7 +1929,7 @@
bb2:=(q*sinh(a*x))/(a*(q^2-p^2)*(p+q*cosh(a*x)))-p/(q^2-p^2)*t1.2
--R Type: Expression
Integer
--E
---S 133
+--S 119
cc1:=aa.1-bb1
--R
--R (5)
@@ -2183,7 +2025,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 134
+--S 120
cc2:=aa.2-bb1
--R
--R (6)
@@ -2271,7 +2113,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 135
+--S 121
cc3:=aa.1-bb2
--R
--R (7)
@@ -2359,7 +2201,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 136 14:582 Axiom cannot simplify this expression
+--S 122 14:582 Axiom cannot simplify this expression
cc4:=aa.2-bb2
--R
--R (8)
@@ -2409,7 +2251,7 @@ $$
<<*>>=
)clear all
---S 137
+--S 123
aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x)
--R
--R
@@ -2477,7 +2319,7 @@ aa:=integrate(1/(p^2-q^2*cosh(a*x)^2),x)
--R Type: Union(List Expression
Integer,...)
--E
---S 138
+--S 124
bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p^2-q^2)))
--R
--R +---------+
@@ -2494,7 +2336,7 @@
bb1:=1/(2*a*p*sqrt(p^2-q^2))*log((p*tanh(a*x)+sqrt(p^2-q^2))/(p*tanh(a*x)-sqrt(p
--R Type: Expression
Integer
--E
---S 139
+--S 125
bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2))
--R
--R p tanh(a x)
@@ -2509,7 +2351,7 @@
bb2:=-1/(a*p*sqrt(q^2-p^2))*atan((p*tanh(a*x))/sqrt(q^2-p^2))
--R Type: Expression
Integer
--E
---S 140
+--S 126
cc1:=aa.1-bb1
--R
--R (4)
@@ -2562,7 +2404,7 @@ cc1:=aa.1-bb1
--R Type: Expression
Integer
--E
---S 141
+--S 127
cc2:=aa.2-bb1
--R
--R (5)
@@ -2599,7 +2441,7 @@ cc2:=aa.2-bb1
--R Type: Expression
Integer
--E
---S 142
+--S 128
cc3:=aa.1-bb2
--R
--R (6)
@@ -2658,7 +2500,7 @@ cc3:=aa.1-bb2
--R Type: Expression
Integer
--E
---S 143 14:583 Axiom cannot simplify this expression
+--S 129 14:583 Axiom cannot simplify this expression
cc4:=aa.2-bb2
--R
--R (7)
@@ -2708,7 +2550,7 @@ $$
<<*>>=
)clear all
---S 144
+--S 130
aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x)
--R
--R
@@ -2754,7 +2596,7 @@ aa:=integrate(1/(p^2+q^2*cosh(a*x)^2),x)
--R Type: Union(Expression
Integer,...)
--E
---S 145
+--S 131
bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p^2+q^2)))
--R
--R +-------+
@@ -2771,7 +2613,7 @@
bb1:=1/(2*a*p*sqrt(p^2+q^2))*log((p*tanh(a*x)+sqrt(p^2+q^2))/(p*tanh(a*x)-sqrt(p
--R Type: Expression
Integer
--E
---S 146
+--S 132
bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2))
--R
--R p tanh(a x)
@@ -2786,7 +2628,7 @@
bb2:=1/(a*p*sqrt(p^2+q^2))*atan((p*tanh(a*x))/sqrt(p^2+q^2))
--R Type: Expression
Integer
--E
---S 147
+--S 133
cc1:=aa-bb1
--R
--R (4)
@@ -2839,7 +2681,7 @@ cc1:=aa-bb1
--R Type: Expression
Integer
--E
---S 148 14:584 Axiom cannot compute this integral
+--S 134 14:584 Axiom cannot compute this integral
cc2:=aa-bb2
--R
--R (5)
@@ -2899,7 +2741,7 @@ $$
<<*>>=
)clear all
---S 149 14:585 Axiom cannot compute this integral
+--S 135 14:585 Axiom cannot compute this integral
aa:=integrate(x^m*cosh(a*x),x)
--R
--R
@@ -2919,7 +2761,7 @@ $$
<<*>>=
)clear all
---S 150 14:586 Axiom cannot compute this integral
+--S 136 14:586 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)^n,x)
--R
--R
@@ -2940,7 +2782,7 @@ $$
<<*>>=
)clear all
---S 151 14:587 Axiom cannot compute this integral
+--S 137 14:587 Axiom cannot compute this integral
aa:=integrate(cosh(a*x)/x^n,x)
--R
--R
@@ -2962,7 +2804,7 @@ $$
<<*>>=
)clear all
---S 152 14:588 Axiom cannot compute this integral
+--S 138 14:588 Axiom cannot compute this integral
aa:=integrate(1/cosh(a*x)^n,x)
--R
--R
@@ -2985,7 +2827,7 @@ $$
<<*>>=
)clear all
---S 153 14:589 Axiom cannot compute this integral
+--S 139 14:589 Axiom cannot compute this integral
aa:=integrate(1/cosh(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum29.input.pamphlet
b/src/input/schaum29.input.pamphlet
index 37e2800..1aff49f 100644
--- a/src/input/schaum29.input.pamphlet
+++ b/src/input/schaum29.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 14
+--S 1
aa:=integrate(sinh(a*x)*cosh(a*x),x)
--R
--R
@@ -28,6 +28,63 @@ aa:=integrate(sinh(a*x)*cosh(a*x),x)
--R 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=sinh(a*x)^2/(2*a)
+--R
+--R 2
+--R sinh(a x)
+--R (2) ----------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R 2 2
+--R - sinh(a x) + cosh(a x)
+--R (3) -------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 4
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 5
+dd:=sinhsqrrule cc
+--R
+--R 2
+--R - cosh(2a x) + 2cosh(a x) + 1
+--R (5) ------------------------------
+--R 8a
+--R Type: Expression
Integer
+--E
+
+--S 6
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 7 14:590 Schaums and Axiom agree
+ee:=coshsqrrule dd
+--R
+--R 1
+--R (7) --
+--R 4a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.591~~~~~$\displaystyle
@@ -38,7 +95,7 @@ $$
<<*>>=
)clear all
---S 2 of 14
+--S 8
aa:=integrate(sinh(p*x)*cosh(q*x),x)
--R
--R
@@ -48,6 +105,35 @@ aa:=integrate(sinh(p*x)*cosh(q*x),x)
--R (q - p )sinh(p x) + (- q + p )cosh(p x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 9
+bb:=(cosh(p+q)*x)/(2*(p+q))+(cosh(p-q)*x)/(2*(p-q))
+--R
+--R (q - p)x cosh(q + p) + (- q - p)x cosh(q - p)
+--R (2) ---------------------------------------------
+--R 2 2
+--R 2q - 2p
+--R Type: Expression
Integer
+--E
+
+--S 10 14:591 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R - 2q sinh(p x)sinh(q x)
+--R +
+--R 2
+--R ((- q + p)x cosh(q + p) + (q + p)x cosh(q - p))sinh(p x)
+--R +
+--R 2p cosh(p x)cosh(q x)
+--R +
+--R 2
+--R ((q - p)x cosh(q + p) + (- q - p)x cosh(q - p))cosh(p x)
+--R /
+--R 2 2 2 2 2 2
+--R (2q - 2p )sinh(p x) + (- 2q + 2p )cosh(p x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.592~~~~~$\displaystyle
@@ -58,7 +144,7 @@ $$
<<*>>=
)clear all
---S 3 of 14
+--S 11
aa:=integrate(sinh(a*x)^n*cosh(a*x),x)
--R
--R
@@ -68,6 +154,29 @@ aa:=integrate(sinh(a*x)^n*cosh(a*x),x)
--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 12
+bb:=sinh(a*x)/((n+1)*a)
+--R
+--R sinh(a x)
+--R (2) ---------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 13 14:592 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R - sinh(a x)sinh(n log(sinh(a x))) - sinh(a x)cosh(n log(sinh(a x)))
+--R +
+--R 3 2
+--R - sinh(a x) + cosh(a x) sinh(a x)
+--R /
+--R 2 2
+--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.593~~~~~$\displaystyle
@@ -78,7 +187,7 @@ $$
<<*>>=
)clear all
---S 4 of 14
+--S 14
aa:=integrate(cosh(a*x)^n*sinh(a*x),x)
--R
--R
@@ -88,6 +197,29 @@ aa:=integrate(cosh(a*x)^n*sinh(a*x),x)
--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 15
+bb:=cosh(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R cosh(a x)
+--R (2) --------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 16 14:593 Axiom cannot simplify this expression
+--R
+--R (3)
+--R - cosh(a x)sinh(n log(cosh(a x))) - cosh(a x)cosh(n log(cosh(a x)))
+--R +
+--R 2 2 n + 1
+--R (- sinh(a x) + cosh(a x) )cosh(a x)
+--R /
+--R 2 2
+--R (a n + a)sinh(a x) + (- a n - a)cosh(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.594~~~~~$\displaystyle
@@ -98,7 +230,7 @@ $$
<<*>>=
)clear all
---S 5 of 14
+--S 17
aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x)
--R
--R
@@ -108,6 +240,22 @@ aa:=integrate(sinh(a*x)^2*cosh(a*x)^2,x)
--R 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 18
+bb:=sinh(4*a*x)/(32*a)-x/8
+--R
+--R sinh(4a x) - 4a x
+--R (2) -----------------
+--R 32a
+--R Type: Expression
Integer
+--E
+
+--S 19 14:594 Schaums and Axiom agree
+cc:=complexNormalize(aa-bb)
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.595~~~~~$\displaystyle
@@ -118,7 +266,7 @@ $$
<<*>>=
)clear all
---S 6 of 14
+--S 20
aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x)
--R
--R
@@ -129,6 +277,67 @@ aa:=integrate(1/(sinh(a*x)*cosh(a*x)),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 21
+bb:=1/a*log(tanh(a*x))
+--R
+--R log(tanh(a x))
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R (3)
+--R 2cosh(a x)
+--R - log(tanh(a x)) - log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2sinh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 23
+dd:=expandLog cc
+--R
+--R - log(tanh(a x)) + log(sinh(a x)) - log(cosh(a x))
+--R (4) --------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 24
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (5) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25
+ee:=tanhrule dd
+--R
+--R sinh(a x)
+--R log(sinh(a x)) - log(---------) - log(cosh(a x))
+--R cosh(a x)
+--R (6) ------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 26 14:595 Schaums and Axiom agree
+ff:=expandLog ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.596~~~~~$\displaystyle
@@ -139,17 +348,52 @@ $$
<<*>>=
)clear all
---S 7 of 14
-aa:=integrate(1/(sinh(a*x)^2*cos(a*x)),x)
---R
+--S 27
+aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)),x)
--R
---R x
---R ++ 1
---I (1) | -------------------- d%R
---R ++ 2
---I cos(%R a)sinh(%R a)
+--R (1)
+--R 2 2
+--R (- 2sinh(a x) - 4cosh(a x)sinh(a x) - 2cosh(a x) + 2)
+--R *
+--R atan(sinh(a x) + cosh(a x))
+--R +
+--R - 2sinh(a x) - 2cosh(a x)
+--R /
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 28
+bb:=-1/a*atan(sinh(a*x)-csch(a*x))/a
+--R
+--R atan(sinh(a x) - csch(a x))
+--R (2) - ---------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 29 14:596 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- 2a sinh(a x) - 4a cosh(a x)sinh(a x) - 2a cosh(a x) + 2a)
+--R *
+--R atan(sinh(a x) + cosh(a x))
+--R +
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1)
+--R *
+--R atan(sinh(a x) - csch(a x))
+--R +
+--R - 2a sinh(a x) - 2a cosh(a x)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.597~~~~~$\displaystyle
@@ -160,7 +404,7 @@ $$
<<*>>=
)clear all
---S 8 of 14
+--S 30
aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x)
--R
--R
@@ -181,6 +425,272 @@ aa:=integrate(1/(sinh(a*x)*cosh(a*x)^2),x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 31
+bb:=sech(a*x)/a+1/a*log(tanh((a*x)/2))
+--R
+--R a x
+--R log(tanh(---)) + sech(a x)
+--R 2
+--R (2) --------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 32
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2 a x
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)log(tanh(---))
+--R 2
+--R +
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2
+--R - sech(a x)sinh(a x) + (- 2cosh(a x)sech(a x) + 2)sinh(a x)
+--R +
+--R 2
+--R (- cosh(a x) - 1)sech(a x) + 2cosh(a x)
+--R /
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 33
+sechrule:=rule(sech(x) == 1/cosh(x))
+--R
+--R 1
+--R (4) sech(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 34
+dd:=sechrule cc
+--R
+--R (5)
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) -
cosh(a x))
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) -
cosh(a x))
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2 3
+--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a
x))
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - sinh(a x) + cosh(a x) - 1
+--R /
+--R 2 2 3
+--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a
cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 35
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (6) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 36
+ee:=tanhrule dd
+--R
+--R (7)
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) -
cosh(a x))
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2 3
+--R (cosh(a x)sinh(a x) + 2cosh(a x) sinh(a x) + cosh(a x) + cosh(a
x))
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2 3
+--R (- cosh(a x)sinh(a x) - 2cosh(a x) sinh(a x) - cosh(a x) -
cosh(a x))
+--R *
+--R a x
+--R sinh(---)
+--R 2
+--R log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R +
+--R 2 2
+--R - sinh(a x) + cosh(a x) - 1
+--R /
+--R 2 2 3
+--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a
cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 37
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R 3 cosh(3x) - 3cosh(x)
+--R (8) cosh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 38
+ff:=coshcuberule ee
+--R
+--R (9)
+--R 2 2
+--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x)
+--R +
+--R - cosh(a x)
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (4cosh(a x)sinh(a x) + 8cosh(a x) sinh(a x) + cosh(3a x) + cosh(a
x))
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - 4cosh(a x)sinh(a x) - 8cosh(a x) sinh(a x) - cosh(3a x)
+--R +
+--R - cosh(a x)
+--R *
+--R a x
+--R sinh(---)
+--R 2
+--R log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R +
+--R 2 2
+--R - 4sinh(a x) + 4cosh(a x) - 4
+--R /
+--R 2 2
+--R 4a cosh(a x)sinh(a x) + 8a cosh(a x) sinh(a x) + a cosh(3a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 39
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (10) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 40
+gg:=coshsqrrule ff
+--R
+--R (11)
+--R 2
+--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a
x)
+--R +
+--R - cosh(a x)
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R 4cosh(a x)sinh(a x) + (4cosh(2a x) + 4)sinh(a x) + cosh(3a x)
+--R +
+--R cosh(a x)
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2
+--R - 4cosh(a x)sinh(a x) + (- 4cosh(2a x) - 4)sinh(a x) - cosh(3a
x)
+--R +
+--R - cosh(a x)
+--R *
+--R a x
+--R sinh(---)
+--R 2
+--R log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R +
+--R 2
+--R - 4sinh(a x) + 2cosh(2a x) - 2
+--R /
+--R 2
+--R 4a cosh(a x)sinh(a x) + (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 41
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (12) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 42
+hh:=sinhsqrrule gg
+--R
+--R (13)
+--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R a x
+--R sinh(---)
+--R 2
+--R - log(---------)
+--R a x
+--R cosh(---)
+--R 2
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 43
+ii:=expandLog hh
+--R
+--R (14)
+--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R a x a x
+--R - log(sinh(---)) + log(cosh(---))
+--R 2 2
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 44 14:597 Schaums and Axiom agree
+jj:=complexNormalize ii
+--R
+--R (15) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.598~~~~~$\displaystyle
@@ -191,7 +701,7 @@ $$
<<*>>=
)clear all
---S 9 of 14
+--S 45
aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x)
--R
--R
@@ -206,6 +716,35 @@ aa:=integrate(1/(sinh(a*x)^2*cosh(a*x)^2),x)
--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 46
+bb:=-(2*coth(2*a*x))/a
+--R
+--R 2coth(2a x)
+--R (2) - -----------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 47 14:598 Axiom cannot simplify this expression
+--R
+--R (3)
+--R 4 3
+--R 2coth(2a x)sinh(a x) + 8cosh(a x)coth(2a x)sinh(a x)
+--R +
+--R 2 2 3
+--R 12cosh(a x) coth(2a x)sinh(a x) + 8cosh(a x) coth(2a x)sinh(a x)
+--R +
+--R 4
+--R (2cosh(a x) - 2)coth(2a x) - 4
+--R /
+--R 4 3 2 2
+--R a sinh(a x) + 4a cosh(a x)sinh(a x) + 6a cosh(a x) sinh(a x)
+--R +
+--R 3 4
+--R 4a cosh(a x) sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.599~~~~~$\displaystyle
@@ -216,7 +755,7 @@ $$
<<*>>=
)clear all
---S 10 of 14
+--S 48
aa:=integrate(sinh(a*x)^2/cosh(a*x),x)
--R
--R
@@ -230,6 +769,28 @@ aa:=integrate(sinh(a*x)^2/cosh(a*x),x)
--R 2a sinh(a x) + 2a cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 49
+bb:=sinh(a*x)/a-1/a*atan(sinh(a*x))
+--R
+--R - atan(sinh(a x)) + sinh(a x)
+--R (2) -----------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 50 14:599 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R (- 4sinh(a x) - 4cosh(a x))atan(sinh(a x) + cosh(a x))
+--R +
+--R 2 2
+--R (2sinh(a x) + 2cosh(a x))atan(sinh(a x)) - sinh(a x) + cosh(a x) -
1
+--R /
+--R 2a sinh(a x) + 2a cosh(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.600~~~~~$\displaystyle
@@ -240,7 +801,7 @@ $$
<<*>>=
)clear all
---S 11 of 14
+--S 51
aa:=integrate(cosh(a*x)^2/sinh(a*x),x)
--R
--R
@@ -256,6 +817,37 @@ aa:=integrate(cosh(a*x)^2/sinh(a*x),x)
--R 2a sinh(a x) + 2a cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 52
+bb:=cosh(a*x)/a+1/a*log(tanh((a*x)/2))
+--R
+--R a x
+--R log(tanh(---)) + cosh(a x)
+--R 2
+--R (2) --------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 53 14:600 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R a x
+--R (- 2sinh(a x) - 2cosh(a x))log(tanh(---))
+--R 2
+--R +
+--R (- 2sinh(a x) - 2cosh(a x))log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R (2sinh(a x) + 2cosh(a x))log(sinh(a x) + cosh(a x) - 1) + sinh(a x)
+--R +
+--R 2
+--R - cosh(a x) + 1
+--R /
+--R 2a sinh(a x) + 2a cosh(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.601~~~~~$\displaystyle
@@ -267,7 +859,7 @@ $$
<<*>>=
)clear all
---S 12 of 14
+--S 54
aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x)
--R
--R
@@ -281,6 +873,87 @@ aa:=integrate(1/(cosh(a*x)*(1+sinh(a*x))),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=1/(2*a)*log((1+sinh(a*x))/cosh(a*x))+1/a*atan(%e^(a*x))
+--R
+--R sinh(a x) + 1 a x
+--R log(-------------) + 2atan(%e )
+--R cosh(a x)
+--R (2) ---------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 56
+cc:=aa-bb
+--R
+--R (3)
+--R sinh(a x) + 1 2cosh(a x)
+--R - log(-------------) - log(- ---------------------)
+--R cosh(a x) sinh(a x) - cosh(a x)
+--R +
+--R - 2sinh(a x) - 2
a x
+--R log(---------------------) + 2atan(sinh(a x) + cosh(a x)) - 2atan(%e
)
+--R sinh(a x) - cosh(a x)
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 57
+dd:=expandLog cc
+--R
+--R a x
+--R atan(sinh(a x) + cosh(a x)) - atan(%e )
+--R (4) -----------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 58
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (5) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 59
+ee:=atanrule dd
+--R
+--R a x
+--R - %e + %i - sinh(a x) - cosh(a x) + %i
+--R %i log(------------) - %i log(----------------------------)
+--R a x sinh(a x) + cosh(a x) + %i
+--R %e + %i
+--R (6) -----------------------------------------------------------
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 60
+ff:=expandLog ee
+--R
+--R (7)
+--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) -
%i)
+--R +
+--R a x a x
+--R - %i log(%e + %i) + %i log(%e - %i)
+--R /
+--R 2a
+--R Type: Expression Complex
Integer
+--E
+
+--S 61 14:601 Schaums and Axiom agree
+gg:=complexNormalize ff
+--R
+--R (8) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.602~~~~~$\displaystyle
@@ -291,7 +964,7 @@ $$
<<*>>=
)clear all
---S 13 of 14
+--S 62
aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x)
--R
--R
@@ -316,6 +989,204 @@ aa:=integrate(1/(sinh(a*x)*(cosh(a*x)+1)),x)
--R 4a cosh(a x) + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 63
+bb:=1/(2*a)*log(tanh((a*x)/2))+1/(2*a*(cosh(a*x)+1))
+--R
+--R a x
+--R (cosh(a x) + 1)log(tanh(---)) + 1
+--R 2
+--R (2) ---------------------------------
+--R 2a cosh(a x) + 2a
+--R Type: Expression
Integer
+--E
+
+--S 64
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R (- cosh(a x) - 1)sinh(a x)
+--R +
+--R 2 3
2
+--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a
x)
+--R +
+--R - 3cosh(a x) - 1
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2
+--R (- cosh(a x) - 1)sinh(a x)
+--R +
+--R 2 3
2
+--R (- 2cosh(a x) - 4cosh(a x) - 2)sinh(a x) - cosh(a x) - 3cosh(a
x)
+--R +
+--R - 3cosh(a x) - 1
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (cosh(a x) + 1)sinh(a x) + (2cosh(a x) + 4cosh(a x) + 2)sinh(a
x)
+--R +
+--R 3 2
+--R cosh(a x) + 3cosh(a x) + 3cosh(a x) + 1
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - sinh(a x) + cosh(a x) - 1
+--R /
+--R 2
+--R (2a cosh(a x) + 2a)sinh(a x)
+--R +
+--R 2 3
+--R (4a cosh(a x) + 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x)
+--R +
+--R 2
+--R 6a cosh(a x) + 6a cosh(a x) + 2a
+--R Type: Expression
Integer
+--E
+
+--S 65
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R 3 cosh(3x) - 3cosh(x)
+--R (4) cosh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 66
+dd:=coshcuberule cc
+--R
+--R (5)
+--R 2
+--R (- 4cosh(a x) - 4)sinh(a x)
+--R +
+--R 2
+--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R - 12cosh(a x) - 9cosh(a x) - 4
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2
+--R (- 4cosh(a x) - 4)sinh(a x)
+--R +
+--R 2
+--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R - 12cosh(a x) - 9cosh(a x) - 4
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2 2
+--R (4cosh(a x) + 4)sinh(a x) + (8cosh(a x) + 16cosh(a x) +
8)sinh(a x)
+--R +
+--R 2
+--R cosh(3a x) + 12cosh(a x) + 9cosh(a x) + 4
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R - 4sinh(a x) + 4cosh(a x) - 4
+--R /
+--R 2
+--R (8a cosh(a x) + 8a)sinh(a x)
+--R +
+--R 2
+--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x)
+--R +
+--R 2
+--R 24a cosh(a x) + 18a cosh(a x) + 8a
+--R Type: Expression
Integer
+--E
+
+--S 67
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (6) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 68
+ee:=sinhsqrrule dd
+--R
+--R (7)
+--R 2
+--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2
+--R (- 8cosh(a x) - 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R (- 2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) - 7cosh(a x) - 2
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R (8cosh(a x) + 16cosh(a x) + 8)sinh(a x) + cosh(3a x)
+--R +
+--R 2
+--R (2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) + 7cosh(a x) + 2
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2
+--R - 2cosh(2a x) + 4cosh(a x) - 2
+--R /
+--R 2
+--R (16a cosh(a x) + 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x)
+--R +
+--R 2
+--R (4a cosh(a x) + 4a)cosh(2a x) + 24a cosh(a x) + 14a cosh(a x) + 4a
+--R Type: Expression
Integer
+--E
+
+--S 69
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (8) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 70
+ff:=coshsqrrule ee
+--R
+--R (9)
+--R a x
+--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1)
+--R 2
+--R +
+--R log(sinh(a x) + cosh(a x) - 1)
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 71 14:602 Schaums and Axiom agree
+gg:=complexNormalize ff
+--R
+--R (10) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.603~~~~~$\displaystyle
@@ -326,7 +1197,7 @@ $$
<<*>>=
)clear all
---S 14 of 14
+--S 72
aa:=integrate(1/(sinh(a*x)*(cosh(a*x)-1)),x)
--R
--R
@@ -352,6 +1223,199 @@ aa:=integrate(1/(sinh(a*x)*(cosh(a*x)-1)),x)
--R Type: Union(Expression
Integer,...)
--E
+--S 73
+bb:=-1/(2*a)*log(tanh((a*x)/2))-1/(2*a*(cosh(a*x)-1))
+--R
+--R a x
+--R (- cosh(a x) + 1)log(tanh(---)) - 1
+--R 2
+--R (2) -----------------------------------
+--R 2a cosh(a x) - 2a
+--R Type: Expression
Integer
+--E
+
+--S 74
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a
x)
+--R +
+--R 3 2
+--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2 2
+--R (cosh(a x) - 1)sinh(a x) + (2cosh(a x) - 4cosh(a x) + 2)sinh(a
x)
+--R +
+--R 3 2
+--R cosh(a x) - 3cosh(a x) + 3cosh(a x) - 1
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R (- cosh(a x) + 1)sinh(a x)
+--R +
+--R 2 3
2
+--R (- 2cosh(a x) + 4cosh(a x) - 2)sinh(a x) - cosh(a x) + 3cosh(a
x)
+--R +
+--R - 3cosh(a x) + 1
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R sinh(a x) - cosh(a x) + 1
+--R /
+--R 2
+--R (2a cosh(a x) - 2a)sinh(a x)
+--R +
+--R 2 3
+--R (4a cosh(a x) - 8a cosh(a x) + 4a)sinh(a x) + 2a cosh(a x)
+--R +
+--R 2
+--R - 6a cosh(a x) + 6a cosh(a x) - 2a
+--R Type: Expression
Integer
+--E
+
+--S 75
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R 3 cosh(3x) - 3cosh(x)
+--R (4) cosh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 76
+dd:=coshcuberule cc
+--R
+--R (5)
+--R 2 2
+--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) +
8)sinh(a x)
+--R +
+--R 2
+--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2 2
+--R (4cosh(a x) - 4)sinh(a x) + (8cosh(a x) - 16cosh(a x) +
8)sinh(a x)
+--R +
+--R 2
+--R cosh(3a x) - 12cosh(a x) + 9cosh(a x) - 4
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R (- 4cosh(a x) + 4)sinh(a x)
+--R +
+--R 2
+--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R 12cosh(a x) - 9cosh(a x) + 4
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2 2
+--R 4sinh(a x) - 4cosh(a x) + 4
+--R /
+--R 2
+--R (8a cosh(a x) - 8a)sinh(a x)
+--R +
+--R 2
+--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x)
+--R +
+--R 2
+--R - 24a cosh(a x) + 18a cosh(a x) - 8a
+--R Type: Expression
Integer
+--E
+
+--S 77
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (6) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 78
+ee:=sinhsqrrule dd
+--R
+--R (7)
+--R 2
+--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x)
+--R +
+--R 2
+--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 2
+--R (8cosh(a x) - 16cosh(a x) + 8)sinh(a x) + cosh(3a x)
+--R +
+--R 2
+--R (2cosh(a x) - 2)cosh(2a x) - 12cosh(a x) + 7cosh(a x) - 2
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 2
+--R (- 8cosh(a x) + 16cosh(a x) - 8)sinh(a x) - cosh(3a x)
+--R +
+--R 2
+--R (- 2cosh(a x) + 2)cosh(2a x) + 12cosh(a x) - 7cosh(a x) + 2
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 2
+--R 2cosh(2a x) - 4cosh(a x) + 2
+--R /
+--R 2
+--R (16a cosh(a x) - 32a cosh(a x) + 16a)sinh(a x) + 2a cosh(3a x)
+--R +
+--R 2
+--R (4a cosh(a x) - 4a)cosh(2a x) - 24a cosh(a x) + 14a cosh(a x) - 4a
+--R Type: Expression
Integer
+--E
+
+--S 79
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (8) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 80
+ff:=coshsqrrule ee
+--R
+--R (9)
+--R a x
+--R log(tanh(---)) + log(sinh(a x) + cosh(a x) + 1)
+--R 2
+--R +
+--R - log(sinh(a x) + cosh(a x) - 1)
+--R /
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 81 14:603 Schaums and Axiom agree
+gg:=complexNormalize ff
+--R
+--R (10) 0
+--R Type: Expression
Integer
+--E
+
)spool
)lisp (bye)
@
diff --git a/src/input/schaum30.input.pamphlet
b/src/input/schaum30.input.pamphlet
index 0e1aaf4..5df48bc 100644
--- a/src/input/schaum30.input.pamphlet
+++ b/src/input/schaum30.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 11
+--S 1
aa:=integrate(tanh(a*x),x)
--R
--R
@@ -29,6 +29,44 @@ aa:=integrate(tanh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/a*log(cosh(a*x))
+--R
+--R log(cosh(a x))
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R 2cosh(a x)
+--R - log(cosh(a x)) + log(- ---------------------) - a x
+--R sinh(a x) - cosh(a x)
+--R (3) -----------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 4
+dd:=expandLog cc
+--R
+--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x
+--R (4) ---------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 5 14:604 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R - log(- 1) + log(- 2)
+--R (5) ---------------------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.605~~~~~$\displaystyle
@@ -39,7 +77,7 @@ $$
<<*>>=
)clear all
---S 2 of 11
+--S 6
aa:=integrate(tanh(a*x)^2,x)
--R
--R
@@ -48,6 +86,42 @@ aa:=integrate(tanh(a*x)^2,x)
--R a cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 7
+bb:=x-tanh(a*x)/a
+--R
+--R - tanh(a x) + a x
+--R (2) -----------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 8
+cc:=aa-bb
+--R
+--R cosh(a x)tanh(a x) - sinh(a x) + cosh(a x)
+--R (3) ------------------------------------------
+--R a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 9
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (4) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 10 14:605 Schaums and Axiom differ by a constant
+dd:=tanhrule cc
+--R
+--R 1
+--R (5) -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.606~~~~~$\displaystyle
@@ -58,7 +132,7 @@ $$
<<*>>=
)clear all
---S 3 of 11
+--S 11
aa:=integrate(tanh(a*x)^3,x)
--R
--R
@@ -92,6 +166,69 @@ aa:=integrate(tanh(a*x)^3,x)
--R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x)
+ a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 12
+bb:=1/a*log(cosh(a*x))-tanh(a*x)^2/(2*a)
+--R
+--R 2
+--R 2log(cosh(a x)) - tanh(a x)
+--R (2) ----------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 13 14:606 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 4 3 2
2
+--R - 2sinh(a x) - 8cosh(a x)sinh(a x) + (- 12cosh(a x) -
4)sinh(a x)
+--R +
+--R 3 4 2
+--R (- 8cosh(a x) - 8cosh(a x))sinh(a x) - 2cosh(a x) - 4cosh(a x)
- 2
+--R *
+--R log(cosh(a x))
+--R +
+--R 4 3 2 2
+--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x)
+--R +
+--R 3 4 2
+--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x)
+ 2
+--R *
+--R 2cosh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 4 3 2 2
+--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) + 2)sinh(a x)
+--R +
+--R 3 4 2
+--R (4cosh(a x) + 4cosh(a x))sinh(a x) + cosh(a x) + 2cosh(a x) +
1
+--R *
+--R 2
+--R tanh(a x)
+--R +
+--R 4 3
+--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x)
+--R +
+--R 2 2
+--R (- 12a x cosh(a x) - 4a x + 4)sinh(a x)
+--R +
+--R 3
4
+--R (- 8a x cosh(a x) + (- 8a x + 8)cosh(a x))sinh(a x) - 2a x cosh(a x)
+--R +
+--R 2
+--R (- 4a x + 4)cosh(a x) - 2a x
+--R /
+--R 4 3 2
2
+--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a
x)
+--R +
+--R 3 4
2
+--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a
x)
+--R +
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.607~~~~~$\displaystyle
@@ -102,7 +239,7 @@ $$
<<*>>=
)clear all
---S 4 of 11
+--S 14
aa:=integrate(tanh(a*x)^n*sech(a*x)^2,x)
--R
--R
@@ -113,6 +250,31 @@ aa:=integrate(tanh(a*x)^n*sech(a*x)^2,x)
--R (a n + a)cosh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 15
+bb:=tanh(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R tanh(a x)
+--R (2) --------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 16 14:607 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R sinh(a x) sinh(a x)
+--R sinh(a x)sinh(n log(---------)) + sinh(a x)cosh(n log(---------))
+--R cosh(a x) cosh(a x)
+--R +
+--R n + 1
+--R - cosh(a x)tanh(a x)
+--R /
+--R (a n + a)cosh(a x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.608~~~~~$\displaystyle
@@ -123,7 +285,7 @@ $$
<<*>>=
)clear all
---S 5 of 11
+--S 17
aa:=integrate(sech(a*x)^2/tanh(a*x),x)
--R
--R
@@ -134,6 +296,63 @@ aa:=integrate(sech(a*x)^2/tanh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 18
+bb:=1/a*log(tanh(a*x))
+--R
+--R log(tanh(a x))
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 19
+cc:=aa-bb
+--R
+--R (3)
+--R 2cosh(a x)
+--R - log(tanh(a x)) - log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2sinh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 20
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (4) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 21
+dd:=tanhrule cc
+--R
+--R (5)
+--R sinh(a x) 2cosh(a x)
+--R - log(---------) - log(- ---------------------)
+--R cosh(a x) sinh(a x) - cosh(a x)
+--R +
+--R 2sinh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 22 14:608 Schaums and Axiom agree
+ee:=expandLog dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.609~~~~~$\displaystyle
@@ -144,7 +363,7 @@ $$
<<*>>=
)clear all
---S 6 of 11
+--S 23
aa:=integrate(1/tanh(a*x),x)
--R
--R
@@ -155,6 +374,44 @@ aa:=integrate(1/tanh(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=1/a*log(sinh(a*x))
+--R
+--R log(sinh(a x))
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R 2sinh(a x)
+--R - log(sinh(a x)) + log(- ---------------------) - a x
+--R sinh(a x) - cosh(a x)
+--R (3) -----------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 26
+dd:=expandLog cc
+--R
+--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x
+--R (4) ---------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 27 14:609 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R - log(- 1) + log(- 2)
+--R (5) ---------------------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.610~~~~~$\displaystyle
@@ -167,7 +424,7 @@ $$
<<*>>=
)clear all
---S 7 of 11
+--S 28 14:610 Axiom cannot compute this integral
aa:=integrate(x*tanh(a*x),x)
--R
--R
@@ -187,7 +444,7 @@ $$
<<*>>=
)clear all
---S 8 of 11
+--S 29
aa:=integrate(x*tanh(a*x)^2,x)
--R
--R
@@ -209,6 +466,159 @@ aa:=integrate(x*tanh(a*x)^2,x)
--R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 30
+bb:=x^2/2-(x*tanh(a*x))/a+1/a^2*log(cosh(a*x))
+--R
+--R 2 2
+--R 2log(cosh(a x)) - 2a x tanh(a x) + a x
+--R (2) ---------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 31
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)log(cosh(a x))
+--R +
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)
+--R *
+--R 2cosh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2 2
+--R (a x sinh(a x) + 2a x cosh(a x)sinh(a x) + a x cosh(a x) + a x)
+--R *
+--R tanh(a x)
+--R +
+--R 2 2
+--R - 2a x sinh(a x) - 4a x cosh(a x)sinh(a x) - 2a x cosh(a x)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 32
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 33
+dd:=sinhsqrrule cc
+--R
+--R (5)
+--R 2
+--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) - 1)log(cosh(a x))
+--R +
+--R 2
+--R (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) + 1)
+--R *
+--R 2cosh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2
+--R (4a x cosh(a x)sinh(a x) + a x cosh(2a x) + 2a x cosh(a x) + a x)
+--R *
+--R tanh(a x)
+--R +
+--R 2
+--R - 8a x cosh(a x)sinh(a x) - 2a x cosh(2a x) - 4a x cosh(a x) + 2a x
+--R /
+--R 2 2 2 2 2
+--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 34
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35
+ee:=coshsqrrule dd
+--R
+--R (7)
+--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(cosh(a x))
+--R +
+--R 2cosh(a x)
+--R (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x)
+--R +
+--R - 4a x cosh(a x)sinh(a x) - 2a x cosh(2a x)
+--R /
+--R 2 2 2
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 36
+ff:=expandLog ee
+--R
+--R (8)
+--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (2a x cosh(a x)sinh(a x) + a x cosh(2a x) + a x)tanh(a x)
+--R +
+--R (2log(- 2) - 4a x)cosh(a x)sinh(a x) + (log(- 2) - 2a x)cosh(2a x)
+--R +
+--R log(- 2)
+--R /
+--R 2 2 2
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 37
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %N sinh(y + x) - %N sinh(y - x)
+--I (9) %N cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 38
+gg:=sinhcoshrule ff
+--R
+--R (10)
+--R (- sinh(2a x) - cosh(2a x) - 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (a x sinh(2a x) + a x cosh(2a x) + a x)tanh(a x)
+--R +
+--R (log(- 2) - 2a x)sinh(2a x) + (log(- 2) - 2a x)cosh(2a x) + log(- 2)
+--R /
+--R 2 2 2
+--R a sinh(2a x) + a cosh(2a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 39 14:611 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R - log(- 1) + log(- 2)
+--R (11) ---------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.612~~~~~$\displaystyle
@@ -220,7 +630,7 @@ $$
<<*>>=
)clear all
---S 9 of 11
+--S 40 14:612 Axiom cannot compute this integral
aa:=integrate(tanh(a*x)/x,x)
--R
--R
@@ -240,7 +650,7 @@ $$
<<*>>=
)clear all
---S 10 of 11
+--S 41
aa:=integrate(1/(p+q*tanh(a*x)),x)
--R
--R
@@ -252,6 +662,54 @@ aa:=integrate(1/(p+q*tanh(a*x)),x)
--R a q - a p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 42
+bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(q*sinh(a*x)+p*cosh(a*x))
+--R
+--R q log(q sinh(a x) + p cosh(a x)) - a p x
+--R (2) ----------------------------------------
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc:=aa-bb
+--R
+--R (3)
+--R - 2q sinh(a x) - 2p
cosh(a x)
+--R - q log(q sinh(a x) + p cosh(a x)) + q
log(-----------------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R - a q x
+--R /
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
+
+--S 44
+dd:=expandLog cc
+--R
+--R (4)
+--R - q log(q sinh(a x) + p cosh(a x)) - q log(sinh(a x) - cosh(a x))
+--R +
+--R q log(- q sinh(a x) - p cosh(a x)) + q log(2) - a q x
+--R /
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
+
+--S 45 14:613 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R q log(2) - 2q log(- 1)
+--R (5) ----------------------
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.614~~~~~$\displaystyle
@@ -262,7 +720,7 @@ $$
<<*>>=
)clear all
---S 11 of 11
+--S 46 14:614 Axiom cannot compute this integral
aa:=integrate(tanh(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum31.input.pamphlet
b/src/input/schaum31.input.pamphlet
index 3e81bf2..343aa2a 100644
--- a/src/input/schaum31.input.pamphlet
+++ b/src/input/schaum31.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 11
+--S 1
aa:=integrate(coth(a*x),x)
--R
--R
@@ -29,6 +29,44 @@ aa:=integrate(coth(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/a*log(sinh(a*x))
+--R
+--R log(sinh(a x))
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R 2sinh(a x)
+--R - log(sinh(a x)) + log(- ---------------------) - a x
+--R sinh(a x) - cosh(a x)
+--R (3) -----------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 4
+dd:=expandLog cc
+--R
+--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x
+--R (4) ---------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 5 14:615 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R - log(- 1) + log(- 2)
+--R (5) ---------------------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.616~~~~~$\displaystyle
@@ -39,7 +77,7 @@ $$
<<*>>=
)clear all
---S 2 of 11
+--S 6
aa:=integrate(coth(a*x)^2,x)
--R
--R
@@ -48,6 +86,33 @@ aa:=integrate(coth(a*x)^2,x)
--R a sinh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 7
+bb:=x-coth(a*x)/a
+--R
+--R - coth(a x) + a x
+--R (2) -----------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 8
+cc:=aa-bb
+--R
+--R (coth(a x) + 1)sinh(a x) - cosh(a x)
+--R (3) ------------------------------------
+--R a sinh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 9 14:616 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R 1
+--R (4) -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.617~~~~~$\displaystyle
@@ -58,7 +123,7 @@ $$
<<*>>=
)clear all
---S 3 of 11
+--S 10
aa:=integrate(coth(a*x)^3,x)
--R
--R
@@ -92,6 +157,69 @@ aa:=integrate(coth(a*x)^3,x)
--R (4a cosh(a x) - 4a cosh(a x))sinh(a x) + a cosh(a x) - 2a cosh(a x)
+ a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 11
+bb:=1/a*log(sinh(a*x)-coth(a*x)^2)/(2*a)
+--R
+--R 2
+--R log(sinh(a x) - coth(a x) )
+--R (2) ---------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 12 14:617 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 4 3 2
2
+--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a
x)
+--R +
+--R 3 4 2
+--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x)
- 1
+--R *
+--R 2
+--R log(sinh(a x) - coth(a x) )
+--R +
+--R 4 3
+--R 2a sinh(a x) + 8a cosh(a x)sinh(a x)
+--R +
+--R 2 2
+--R (12a cosh(a x) - 4a)sinh(a x)
+--R +
+--R 3 4
+--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x)
+--R +
+--R 2
+--R - 4a cosh(a x) + 2a
+--R *
+--R 2sinh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2 4 2 3
+--R - 2a x sinh(a x) - 8a x cosh(a x)sinh(a x)
+--R +
+--R 2 2 2 2
+--R (- 12a x cosh(a x) + 4a x - 4a)sinh(a x)
+--R +
+--R 2 3 2 2 4
+--R (- 8a x cosh(a x) + (8a x - 8a)cosh(a x))sinh(a x) - 2a x cosh(a x)
+--R +
+--R 2 2 2
+--R (4a x - 4a)cosh(a x) - 2a x
+--R /
+--R 2 4 2 3 2 2 2
2
+--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a
)sinh(a x)
+--R +
+--R 2 3 2 2 4 2
2
+--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a
x)
+--R +
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.618~~~~~$\displaystyle
@@ -102,7 +230,7 @@ $$
<<*>>=
)clear all
---S 4 of 11
+--S 13
aa:=integrate(coth(a*x)^n*csch(a*x)^2,x)
--R
--R
@@ -113,6 +241,53 @@ aa:=integrate(coth(a*x)^n*csch(a*x)^2,x)
--R (a n + a)sinh(a x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 14
+bb:=-coth(a*x)^(n+1)/((n+1)*a)
+--R
+--R n + 1
+--R coth(a x)
+--R (2) - --------------
+--R a n + a
+--R Type: Expression
Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R (3)
+--R cosh(a x) cosh(a x)
+--R - cosh(a x)sinh(n log(---------)) - cosh(a x)cosh(n log(---------))
+--R sinh(a x) sinh(a x)
+--R +
+--R n + 1
+--R sinh(a x)coth(a x)
+--R /
+--R (a n + a)sinh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 16
+dd:=expandLog cc
+--R
+--R (4)
+--R cosh(a x)sinh(n log(sinh(a x)) - n log(cosh(a x)))
+--R +
+--R - cosh(a x)cosh(n log(sinh(a x)) - n log(cosh(a x)))
+--R +
+--R n + 1
+--R sinh(a x)coth(a x)
+--R /
+--R (a n + a)sinh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 17 14:618 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.619~~~~~$\displaystyle
@@ -123,7 +298,7 @@ $$
<<*>>=
)clear all
---S 5 of 11
+--S 18
aa:=integrate(csch(a*x)^2/coth(a*x),x)
--R
--R
@@ -134,6 +309,43 @@ aa:=integrate(csch(a*x)^2/coth(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 19
+bb:=-1/a*log(coth(a*x))
+--R
+--R log(coth(a x))
+--R (2) - --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 20
+cc:=aa-bb
+--R
+--R (3)
+--R 2cosh(a x) 2sinh(a x)
+--R log(coth(a x)) - log(- ---------------------) + log(-
---------------------)
+--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a
x)
+--R
----------------------------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 21
+dd:=expandLog cc
+--R
+--R log(sinh(a x)) + log(coth(a x)) - log(cosh(a x))
+--R (4) ------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 22 14:619 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.620~~~~~$\displaystyle
@@ -144,7 +356,7 @@ $$
<<*>>=
)clear all
---S 6 of 11
+--S 23
aa:=integrate(1/coth(a*x),x)
--R
--R
@@ -155,6 +367,44 @@ aa:=integrate(1/coth(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=1/a*log(cosh(a*x))
+--R
+--R log(cosh(a x))
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R 2cosh(a x)
+--R - log(cosh(a x)) + log(- ---------------------) - a x
+--R sinh(a x) - cosh(a x)
+--R (3) -----------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 26
+dd:=expandLog cc
+--R
+--R - log(sinh(a x) - cosh(a x)) + log(- 2) - a x
+--R (4) ---------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 27 14:620 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R - log(- 1) + log(- 2)
+--R (5) ---------------------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.621~~~~~$\displaystyle
@@ -167,7 +417,7 @@ $$
<<*>>=
)clear all
---S 7 of 11
+--S 28 14:621 Axiom cannot compute this integral
aa:=integrate(x*coth(a*x),x)
--R
--R
@@ -187,7 +437,7 @@ $$
<<*>>=
)clear all
---S 8 of 11
+--S 29
aa:=integrate(x*coth(a*x)^2,x)
--R
--R
@@ -209,6 +459,157 @@ aa:=integrate(x*coth(a*x)^2,x)
--R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 30
+bb:=x^2/2-(x*coth(a*x)/a)+1/a^2*log(sinh(a*x))
+--R
+--R 2 2
+--R 2log(sinh(a x)) - 2a x coth(a x) + a x
+--R (2) ---------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 31
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x))
+--R +
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1)
+--R *
+--R 2sinh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2
+--R (a x coth(a x) - 2a x)sinh(a x)
+--R +
+--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x)
+--R +
+--R 2 2
+--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 32
+dd:=expandLog cc
+--R
+--R (4)
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)
+--R *
+--R log(sinh(a x) - cosh(a x))
+--R +
+--R 2
+--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x)
+--R +
+--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x)
+--R +
+--R 2 2
+--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) -
log(- 2)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 33
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (5) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 34
+ee:=sinhsqrrule dd
+--R
+--R (6)
+--R 2
+--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3)
+--R *
+--R log(sinh(a x) - cosh(a x))
+--R +
+--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x)
+--R +
+--R 2
+--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x)
+--R +
+--R 2
+--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(-
2)
+--R +
+--R 2a x
+--R /
+--R 2 2 2 2 2
+--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a
+--R Type: Expression
Integer
+--E
+
+--S 35
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (7) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 36
+ff:=coshsqrrule ee
+--R
+--R (8)
+--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x)
+--R +
+--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) -
log(- 2)
+--R /
+--R 2 2 2
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 37
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %L sinh(y + x) - %L sinh(y - x)
+--I (9) %L cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 38
+gg:=sinhcoshrule ff
+--R
+--R (10)
+--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x)
+--R +
+--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) -
log(- 2)
+--R /
+--R 2 2 2
+--R a sinh(2a x) + a cosh(2a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 39 14:622 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R - log(- 1) + log(- 2)
+--R (11) ---------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.623~~~~~$\displaystyle
@@ -220,7 +621,7 @@ $$
<<*>>=
)clear all
---S 9 of 11
+--S 40 14:623 Axiom cannot compute this integral
aa:=integrate(coth(a*x)/x,x)
--R
--R
@@ -240,7 +641,7 @@ $$
<<*>>=
)clear all
---S 10 of 11
+--S 41
aa:=integrate(1/(p+q*coth(a*x)),x)
--R
--R
@@ -252,6 +653,54 @@ aa:=integrate(1/(p+q*coth(a*x)),x)
--R a q - a p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 42
+bb:=(p*x)/(p^2-q^2)-q/(a*(p^2-q^2))*log(p*sinh(a*x)+q*cosh(a*x))
+--R
+--R q log(p sinh(a x) + q cosh(a x)) - a p x
+--R (2) ----------------------------------------
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc:=aa-bb
+--R
+--R (3)
+--R - 2p sinh(a x) - 2q
cosh(a x)
+--R - q log(p sinh(a x) + q cosh(a x)) + q
log(-----------------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R - a q x
+--R /
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
+
+--S 44
+dd:=expandLog cc
+--R
+--R (4)
+--R - q log(p sinh(a x) + q cosh(a x)) - q log(sinh(a x) - cosh(a x))
+--R +
+--R q log(- p sinh(a x) - q cosh(a x)) + q log(2) - a q x
+--R /
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
+
+--S 45 14:624 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R q log(2) - 2q log(- 1)
+--R (5) ----------------------
+--R 2 2
+--R a q - a p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.625~~~~~$\displaystyle
@@ -262,7 +711,7 @@ $$
<<*>>=
)clear all
---S 11 of 11
+--S 46 14:625 Axiom cannot compute this integral
aa:=integrate(coth(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum32.input.pamphlet
b/src/input/schaum32.input.pamphlet
index 085ddf0..e5b2409 100644
--- a/src/input/schaum32.input.pamphlet
+++ b/src/input/schaum32.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 10
+--S 1
aa:=integrate(sech(a*x),x)
--R
--R
@@ -27,6 +27,70 @@ aa:=integrate(sech(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=2/a*atan(%e^(a*x))
+--R
+--R a x
+--R 2atan(%e )
+--R (2) ------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R a x
+--R 2atan(sinh(a x) + cosh(a x)) - 2atan(%e )
+--R (3) -------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 4
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (4) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 5
+dd:=atanrule cc
+--R
+--R a x
+--R - %e + %i - sinh(a x) - cosh(a x) + %i
+--R %i log(------------) - %i log(----------------------------)
+--R a x sinh(a x) + cosh(a x) + %i
+--R %e + %i
+--R (5) -----------------------------------------------------------
+--R a
+--R Type: Expression Complex
Integer
+--E
+
+--S 6
+ee:=expandLog dd
+--R
+--R (6)
+--R %i log(sinh(a x) + cosh(a x) + %i) - %i log(sinh(a x) + cosh(a x) -
%i)
+--R +
+--R a x a x
+--R - %i log(%e + %i) + %i log(%e - %i)
+--R /
+--R a
+--R Type: Expression Complex
Integer
+--E
+
+--S 7 14:626 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R (7) 0
+--R Type: Expression Complex
Integer
+--E
@
\section{\cite{1}:14.627~~~~~$\displaystyle
@@ -37,7 +101,7 @@ $$
<<*>>=
)clear all
---S 2 of 10
+--S 8
aa:=integrate(sech(a*x)^2,x)
--R
--R
@@ -47,6 +111,91 @@ aa:=integrate(sech(a*x)^2,x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 9
+bb:=tanh(a*x)/a
+--R
+--R tanh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 10
+cc:=aa-bb
+--R
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2
+--R (3) ------------------------------------------------------------------
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 11
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (4) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 12
+dd:=sinhsqrrule cc
+--R
+--R 2
+--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) - 1)tanh(a x) - 4
+--R (5) -------------------------------------------------------------------
+--R 2
+--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 13
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (6) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 14
+ee:=coshsqrrule dd
+--R
+--R (- 2cosh(a x)sinh(a x) - cosh(2a x) - 1)tanh(a x) - 2
+--R (7) -----------------------------------------------------
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 15
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %L sinh(y + x) - %L sinh(y - x)
+--I (8) %L cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 16
+ff:=sinhcoshrule ee
+--R
+--R (- sinh(2a x) - cosh(2a x) - 1)tanh(a x) - 2
+--R (9) --------------------------------------------
+--R a sinh(2a x) + a cosh(2a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 17 14:627 Schaums and Axiom differ by a constant
+gg:=complexNormalize ff
+--R
+--R 1
+--R (10) - -
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.628~~~~~$\displaystyle
@@ -57,7 +206,7 @@ $$
<<*>>=
)clear all
---S 3 of 10
+--S 18
aa:=integrate(sech(a*x)^3,x)
--R
--R
@@ -83,6 +232,65 @@ aa:=integrate(sech(a*x)^3,x)
--R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x)
+ a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 19
+bb:=(sech(a*x)*tanh(a*x))/(2*a)+1/(2*a)*atan(sinh(a*x))
+--R
+--R atan(sinh(a x)) + sech(a x)tanh(a x)
+--R (2) ------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 20 14:628 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 4 3 2 2
+--R 2sinh(a x) + 8cosh(a x)sinh(a x) + (12cosh(a x) + 4)sinh(a x)
+--R +
+--R 3 4 2
+--R (8cosh(a x) + 8cosh(a x))sinh(a x) + 2cosh(a x) + 4cosh(a x)
+ 2
+--R *
+--R atan(sinh(a x) + cosh(a x))
+--R +
+--R 4 3 2
2
+--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) - 2)sinh(a
x)
+--R +
+--R 3 4 2
+--R (- 4cosh(a x) - 4cosh(a x))sinh(a x) - cosh(a x) - 2cosh(a x)
- 1
+--R *
+--R atan(sinh(a x))
+--R +
+--R 4 3
+--R - sech(a x)sinh(a x) - 4cosh(a x)sech(a x)sinh(a x)
+--R +
+--R 2 2
+--R (- 6cosh(a x) - 2)sech(a x)sinh(a x)
+--R +
+--R 3
+--R (- 4cosh(a x) - 4cosh(a x))sech(a x)sinh(a x)
+--R +
+--R 4 2
+--R (- cosh(a x) - 2cosh(a x) - 1)sech(a x)
+--R *
+--R tanh(a x)
+--R +
+--R 3 2 2
+--R 2sinh(a x) + 6cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x)
+--R +
+--R 3
+--R 2cosh(a x) - 2cosh(a x)
+--R /
+--R 4 3 2
2
+--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) + 4a)sinh(a
x)
+--R +
+--R 3 4
2
+--R (8a cosh(a x) + 8a cosh(a x))sinh(a x) + 2a cosh(a x) + 4a cosh(a
x)
+--R +
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.629~~~~~$\displaystyle
@@ -93,7 +301,7 @@ $$
<<*>>=
)clear all
---S 4 of 10
+--S 21
aa:=integrate(sech(a*x)^n*tanh(a*x),x)
--R
--R
@@ -111,6 +319,100 @@ aa:=integrate(sech(a*x)^n*tanh(a*x),x)
--R a n
--R Type: Union(Expression
Integer,...)
--E
+
+--S 22
+bb:=-sech(a*x)^n/(n*a)
+--R
+--R n
+--R sech(a x)
+--R (2) - ----------
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 23
+cc:=aa-bb
+--R
+--R (3)
+--R 2sinh(a x) + 2cosh(a x)
+--R - sinh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1
+--R +
+--R 2sinh(a x) + 2cosh(a x)
+--R - cosh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1
+--R +
+--R n
+--R sech(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 24
+sechrule:=rule(sech(x) == 1/cosh(x))
+--R
+--R 1
+--R (4) sech(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 25
+dd:=sechrule cc
+--R
+--R (5)
+--R 2sinh(a x) + 2cosh(a x)
+--R - sinh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1
+--R +
+--R 2sinh(a x) + 2cosh(a x)
+--R - cosh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1
+--R +
+--R 1 n
+--R (---------)
+--R cosh(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 26
+ee:=expandLog dd
+--R
+--R (6)
+--R sinh
+--R 2 2
+--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R -
+--R cosh
+--R 2 2
+--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R 1 n
+--R (---------)
+--R cosh(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 27 14:629 Schaums and Axiom agree
+ff:=complexNormalize ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.630~~~~~$\displaystyle
@@ -121,7 +423,7 @@ $$
<<*>>=
)clear all
---S 5 of 10
+--S 28
aa:=integrate(1/sech(a*x),x)
--R
--R
@@ -130,6 +432,22 @@ aa:=integrate(1/sech(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 29
+bb:=sinh(a*x)/a
+--R
+--R sinh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 30 14:630 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.631~~~~~$\displaystyle
@@ -142,7 +460,7 @@ $$
<<*>>=
)clear all
---S 6 of 10
+--S 31 14:631 Axiom cannot compute this integral
aa:=integrate(x*sech(a*x),x)
--R
--R
@@ -162,7 +480,7 @@ $$
<<*>>=
)clear all
---S 7 of 10
+--S 32
aa:=integrate(x*sech(a*x)^2,x)
--R
--R
@@ -181,6 +499,161 @@ aa:=integrate(x*sech(a*x)^2,x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 33
+bb:=(x*tanh(a*x))/a-1/a^2*log(cosh(a*x))
+--R
+--R - log(cosh(a x)) + a x tanh(a x)
+--R (2) --------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 34
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)log(cosh(a x))
+--R +
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)
+--R *
+--R 2cosh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2 2
+--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x)
+--R *
+--R tanh(a x)
+--R +
+--R 2 2
+--R 2a x sinh(a x) + 4a x cosh(a x)sinh(a x) + 2a x cosh(a x)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 35
+dd:=expandLog cc
+--R
+--R (4)
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1)
+--R *
+--R log(sinh(a x) - cosh(a x))
+--R +
+--R 2 2
+--R (- a x sinh(a x) - 2a x cosh(a x)sinh(a x) - a x cosh(a x) - a x)
+--R *
+--R tanh(a x)
+--R +
+--R 2
+--R (- log(- 2) + 2a x)sinh(a x) + (- 2log(- 2) + 4a x)cosh(a x)sinh(a
x)
+--R +
+--R 2
+--R (- log(- 2) + 2a x)cosh(a x) - log(- 2)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 36
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (5) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 37
+ee:=sinhsqrrule dd
+--R
+--R (6)
+--R 2
+--R (4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) + 1)
+--R *
+--R log(sinh(a x) - cosh(a x))
+--R +
+--R 2
+--R (- 4a x cosh(a x)sinh(a x) - a x cosh(2a x) - 2a x cosh(a x) - a
x)
+--R *
+--R tanh(a x)
+--R +
+--R (- 4log(- 2) + 8a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a
x)
+--R +
+--R 2
+--R (- 2log(- 2) + 4a x)cosh(a x) - log(- 2) - 2a x
+--R /
+--R 2 2 2 2 2
+--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 38
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (7) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 39
+ff:=coshsqrrule ee
+--R
+--R (8)
+--R (2cosh(a x)sinh(a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (- 2a x cosh(a x)sinh(a x) - a x cosh(2a x) - a x)tanh(a x)
+--R +
+--R (- 2log(- 2) + 4a x)cosh(a x)sinh(a x) + (- log(- 2) + 2a x)cosh(2a
x)
+--R +
+--R - log(- 2)
+--R /
+--R 2 2 2
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 40
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %P sinh(y + x) - %P sinh(y - x)
+--I (9) %P cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 41
+gg:=sinhcoshrule ff
+--R
+--R (10)
+--R (sinh(2a x) + cosh(2a x) + 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (- a x sinh(2a x) - a x cosh(2a x) - a x)tanh(a x)
+--R +
+--R (- log(- 2) + 2a x)sinh(2a x) + (- log(- 2) + 2a x)cosh(2a x) -
log(- 2)
+--R /
+--R 2 2 2
+--R a sinh(2a x) + a cosh(2a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 42 14:632 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R log(- 1) - log(- 2)
+--R (11) -------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.633~~~~~$\displaystyle
@@ -192,7 +665,7 @@ $$
<<*>>=
)clear all
---S 8 of 10
+--S 43 14:633 Axiom cannot compute this integral
aa:=integrate(sech(a*x)/x,x)
--R
--R
@@ -212,7 +685,7 @@ $$
<<*>>=
)clear all
---S 9 of 10
+--S 44
aa:=integrate(1/(q+p*sech(a*x)),x)
--R
--R
@@ -259,6 +732,238 @@ aa:=integrate(1/(q+p*sech(a*x)),x)
--R a q\|q - p
--R Type: Union(List Expression
Integer,...)
--E
+
+--S 45
+t1:=integrate(1/(p+q*cosh(a*x)),x)
+--R
+--R (2)
+--R [
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a
x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R /
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R ,
+--R +-------+
+--R | 2 2
+--R (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R 2atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R ------------------------------------------------]
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R Type: Union(List Expression
Integer,...)
+--E
+
+--S 46
+bb1:=x/q-p/q*t1.1
+--R
+--R (3)
+--R -
+--R p
+--R *
+--R log
+--R 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x)
+--R +
+--R 2 2 2 2
+--R q cosh(a x) + 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2
3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q -
2p
+--R /
+--R 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R +---------+
+--R | 2 2
+--R a x\|- q + p
+--R /
+--R +---------+
+--R | 2 2
+--R a q\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 47
+bb2:=x/q-p/q*t1.2
+--R
+--R +-------+
+--R | 2 2 +-------+
+--R (q sinh(a x) + q cosh(a x) + p)\|q - p | 2 2
+--R - 2p atan(-----------------------------------------) + a x\|q - p
+--R 2 2
+--R q - p
+--R (4) --------------------------------------------------------------------
+--R +-------+
+--R | 2 2
+--R a q\|q - p
+--R Type: Expression
Integer
+--E
+
+--S 48
+cc1:=aa.1-bb1
+--R
+--R (5)
+--R p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q +
2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R /
+--R +---------+
+--R | 2 2
+--R a q\|- q + p
+--R Type: Expression
Integer
+--E
+
+--S 49
+cc2:=aa.2-bb1
+--R
+--R (6)
+--R +-------+
+--R | 2 2
+--R p\|q - p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q - 2p q)sinh(a x) + (2q - 2p q)cosh(a x) + 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R - 2p\|- q + p atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 50
+cc3:=aa.1-bb2
+--R
+--R (7)
+--R +-------+
+--R | 2 2
+--R p\|q - p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) - q + 2p
+--R *
+--R +---------+
+--R | 2 2
+--R \|- q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q +
2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) + q
+--R +
+--R +-------+
+--R +---------+ | 2 2
+--R | 2 2 (q sinh(a x) + q cosh(a x) + p)\|q - p
+--R 2p\|- q + p atan(-----------------------------------------)
+--R 2 2
+--R q - p
+--R /
+--R +---------+ +-------+
+--R | 2 2 | 2 2
+--R a q\|- q + p \|q - p
+--R Type: Expression
Integer
+--E
+
+--S 51 14:634 Schaums and Axiom agree
+cc4:=aa.2-bb2
+--R
+--R (8) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.635~~~~~$\displaystyle
@@ -270,7 +975,7 @@ $$
<<*>>=
)clear all
---S 10 of 10
+--S 52 14:635 Axiom cannot compute this integral
aa:=integrate(sech(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum33.input.pamphlet
b/src/input/schaum33.input.pamphlet
index c5ee6a2..74e0c8b 100644
--- a/src/input/schaum33.input.pamphlet
+++ b/src/input/schaum33.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 10
+--S 1
aa:=integrate(csch(a*x),x)
--R
--R
@@ -27,6 +27,38 @@ aa:=integrate(csch(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/a*log(tanh((a*x)/2))
+--R
+--R a x
+--R log(tanh(---))
+--R 2
+--R (2) --------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R (3)
+--R a x
+--R - log(tanh(---)) - log(sinh(a x) + cosh(a x) + 1)
+--R 2
+--R +
+--R log(sinh(a x) + cosh(a x) - 1)
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 4 14:636 Schaums and Axiom agree
+dd:=complexNormalize cc
+--R
+--R (4) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.637~~~~~$\displaystyle
@@ -37,7 +69,7 @@ $$
<<*>>=
)clear all
---S 2 of 10
+--S 5
aa:=integrate(csch(a*x)^2,x)
--R
--R
@@ -47,6 +79,30 @@ aa:=integrate(csch(a*x)^2,x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 6
+bb:=-coth(a*x)/a
+--R
+--R coth(a x)
+--R (2) - ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 7 14:637 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R coth(a x)sinh(a x) + 2cosh(a x)coth(a x)sinh(a x)
+--R +
+--R 2
+--R (cosh(a x) - 1)coth(a x) - 2
+--R /
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.638~~~~~$\displaystyle
@@ -57,7 +113,7 @@ $$
<<*>>=
)clear all
---S 3 of 10
+--S 8
aa:=integrate(csch(a*x)^3,x)
--R
--R
@@ -93,6 +149,72 @@ aa:=integrate(csch(a*x)^3,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 9
+bb:=-(csch(a*x)*coth(a*x))/(2*a)-1/(2*a)*log(tanh((a*x)/2))
+--R
+--R a x
+--R - log(tanh(---)) - coth(a x)csch(a x)
+--R 2
+--R (2) -------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 10 14:638 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R 4 3 2 2
+--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x)
+--R +
+--R 3 4 2
+--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) +
1
+--R *
+--R a x
+--R log(tanh(---))
+--R 2
+--R +
+--R 4 3 2 2
+--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x)
+--R +
+--R 3 4 2
+--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) +
1
+--R *
+--R log(sinh(a x) + cosh(a x) + 1)
+--R +
+--R 4 3 2
2
+--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a
x)
+--R +
+--R 3 4 2
+--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x)
- 1
+--R *
+--R log(sinh(a x) + cosh(a x) - 1)
+--R +
+--R 4
+--R coth(a x)csch(a x)sinh(a x)
+--R +
+--R 3
+--R (4cosh(a x)coth(a x)csch(a x) - 2)sinh(a x)
+--R +
+--R 2 2
+--R ((6cosh(a x) - 2)coth(a x)csch(a x) - 6cosh(a x))sinh(a x)
+--R +
+--R 3 2
+--R ((4cosh(a x) - 4cosh(a x))coth(a x)csch(a x) - 6cosh(a x) -
2)sinh(a x)
+--R +
+--R 4 2 3
+--R (cosh(a x) - 2cosh(a x) + 1)coth(a x)csch(a x) - 2cosh(a x) -
2cosh(a x)
+--R /
+--R 4 3 2
2
+--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a)sinh(a
x)
+--R +
+--R 3 4
2
+--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a
x)
+--R +
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.639~~~~~$\displaystyle
@@ -103,7 +225,7 @@ $$
<<*>>=
)clear all
---S 4 of 10
+--S 11
aa:=integrate(csch(a*x)^n*coth(a*x),x)
--R
--R
@@ -121,6 +243,202 @@ aa:=integrate(csch(a*x)^n*coth(a*x),x)
--R a n
--R Type: Union(Expression
Integer,...)
--E
+
+--S 12
+bb:=-csch(a*x)^n/(n*a)
+--R
+--R n
+--R csch(a x)
+--R (2) - ----------
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 13
+cc:=aa-bb
+--R
+--R (3)
+--R 2sinh(a x) + 2cosh(a x)
+--R - sinh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1
+--R +
+--R 2sinh(a x) + 2cosh(a x)
+--R - cosh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1
+--R +
+--R n
+--R csch(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 14
+cschrule:=rule(csch(x) == 1/sinh(x))
+--R
+--R 1
+--R (4) csch(x) == -------
+--R sinh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 15
+dd:=cschrule cc
+--R
+--R (5)
+--R 2sinh(a x) + 2cosh(a x)
+--R - sinh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1
+--R +
+--R 2sinh(a x) + 2cosh(a x)
+--R - cosh(n log(-------------------------------------------------))
+--R 2 2
+--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1
+--R +
+--R 1 n
+--R (---------)
+--R sinh(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 16
+ee:=expandLog dd
+--R
+--R (6)
+--R sinh
+--R 2 2
+--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1)
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R -
+--R cosh
+--R 2 2
+--R n log(sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1)
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R 1 n
+--R (---------)
+--R sinh(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 17
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (7) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 18
+ff:=sinhsqrrule ee
+--R
+--R (8)
+--R sinh
+--R 2
+--R 4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) - 3
+--R n log(--------------------------------------------------)
+--R 2
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R -
+--R cosh
+--R 2
+--R 4cosh(a x)sinh(a x) + cosh(2a x) + 2cosh(a x) - 3
+--R n log(--------------------------------------------------)
+--R 2
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R 1 n
+--R (---------)
+--R sinh(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 19
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (9) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 20
+gg:=coshsqrrule ff
+--R
+--R (10)
+--R sinh
+--R n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1)
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R -
+--R cosh
+--R n log(2cosh(a x)sinh(a x) + cosh(2a x) - 1)
+--R +
+--R - n log(sinh(a x) + cosh(a x)) - n log(2)
+--R +
+--R 1 n
+--R (---------)
+--R sinh(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 21
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %O sinh(y + x) - %O sinh(y - x)
+--I (11) %O cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 22
+hh:=sinhcoshrule gg
+--R
+--R (12)
+--R sinh
+--R n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) + cosh(a
x))
+--R +
+--R - n log(2)
+--R +
+--R -
+--R cosh
+--R n log(sinh(2a x) + cosh(2a x) - 1) - n log(sinh(a x) +
cosh(a x))
+--R +
+--R - n log(2)
+--R +
+--R 1 n
+--R (---------)
+--R sinh(a x)
+--R /
+--R a n
+--R Type: Expression
Integer
+--E
+
+--S 23 14:639 Schaums and Axiom agree
+ii:=complexNormalize hh
+--R
+--R (13) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.640~~~~~$\displaystyle
@@ -131,7 +449,7 @@ $$
<<*>>=
)clear all
---S 5 of 10
+--S 24
aa:=integrate(1/csch(a*x),x)
--R
--R
@@ -140,6 +458,22 @@ aa:=integrate(1/csch(a*x),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 25
+bb:=1/a*cosh(a*x)
+--R
+--R cosh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 26 14:640 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.641~~~~~$\displaystyle
@@ -152,7 +486,7 @@ $$
<<*>>=
)clear all
---S 6 of 10
+--S 27 14:641 Axiom cannot compute this integral
aa:=integrate(x*csch(a*x),x)
--R
--R
@@ -172,7 +506,7 @@ $$
<<*>>=
)clear all
---S 7 of 10
+--S 28
aa:=integrate(x*csch(a*x)^2,x)
--R
--R
@@ -191,6 +525,156 @@ aa:=integrate(x*csch(a*x)^2,x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 29
+bb:=-(x*coth(a*x))/a+1/a^2*log(sinh(a*x))
+--R
+--R log(sinh(a x)) - a x coth(a x)
+--R (2) ------------------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 30
+cc:=aa-bb
+--R
+--R (3)
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)log(sinh(a x))
+--R +
+--R 2 2
+--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1)
+--R *
+--R 2sinh(a x)
+--R log(- ---------------------)
+--R sinh(a x) - cosh(a x)
+--R +
+--R 2
+--R (a x coth(a x) - 2a x)sinh(a x)
+--R +
+--R (2a x cosh(a x)coth(a x) - 4a x cosh(a x))sinh(a x)
+--R +
+--R 2 2
+--R (a x cosh(a x) - a x)coth(a x) - 2a x cosh(a x)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 31
+dd:=expandLog cc
+--R
+--R (4)
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) + 1)
+--R *
+--R log(sinh(a x) - cosh(a x))
+--R +
+--R 2
+--R (a x coth(a x) + log(- 2) - 2a x)sinh(a x)
+--R +
+--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x)
+--R +
+--R 2 2
+--R (a x cosh(a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(a x) -
log(- 2)
+--R /
+--R 2 2 2 2 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 32
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (5) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 33
+ee:=sinhsqrrule dd
+--R
+--R (6)
+--R 2
+--R (- 4cosh(a x)sinh(a x) - cosh(2a x) - 2cosh(a x) + 3)
+--R *
+--R log(sinh(a x) - cosh(a x))
+--R +
+--R (4a x cosh(a x)coth(a x) + (4log(- 2) - 8a x)cosh(a x))sinh(a x)
+--R +
+--R 2
+--R (a x cosh(2a x) + 2a x cosh(a x) - 3a x)coth(a x)
+--R +
+--R 2
+--R (log(- 2) - 2a x)cosh(2a x) + (2log(- 2) - 4a x)cosh(a x) - 3log(-
2)
+--R +
+--R 2a x
+--R /
+--R 2 2 2 2 2
+--R 4a cosh(a x)sinh(a x) + a cosh(2a x) + 2a cosh(a x) - 3a
+--R Type: Expression
Integer
+--E
+
+--S 34
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (7) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35
+ff:=coshsqrrule ee
+--R
+--R (8)
+--R (- 2cosh(a x)sinh(a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (2a x cosh(a x)coth(a x) + (2log(- 2) - 4a x)cosh(a x))sinh(a x)
+--R +
+--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) -
log(- 2)
+--R /
+--R 2 2 2
+--R 2a cosh(a x)sinh(a x) + a cosh(2a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 36
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %P sinh(y + x) - %P sinh(y - x)
+--I (9) %P cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 37
+gg:=sinhcoshrule ff
+--R
+--R (10)
+--R (- sinh(2a x) - cosh(2a x) + 1)log(sinh(a x) - cosh(a x))
+--R +
+--R (a x coth(a x) + log(- 2) - 2a x)sinh(2a x)
+--R +
+--R (a x cosh(2a x) - a x)coth(a x) + (log(- 2) - 2a x)cosh(2a x) -
log(- 2)
+--R /
+--R 2 2 2
+--R a sinh(2a x) + a cosh(2a x) - a
+--R Type: Expression
Integer
+--E
+
+--S 38 14:642 Axiom cannot simplify this expression
+hh:=complexNormalize gg
+--R
+--R - log(- 1) + log(- 2)
+--R (11) ---------------------
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.643~~~~~$\displaystyle
@@ -202,7 +686,7 @@ $$
<<*>>=
)clear all
---S 8 of 10
+--S 39 14:643 Axiom cannot compute this integral
aa:=integrate(csch(a*x)/x,x)
--R
--R
@@ -222,7 +706,7 @@ $$
<<*>>=
)clear all
---S 9 of 10
+--S 40
aa:=integrate(1/(q+p*csch(a*x)),x)
--R
--R
@@ -257,6 +741,251 @@ aa:=integrate(1/(q+p*csch(a*x)),x)
--R a q\|q + p
--R Type: Union(Expression
Integer,...)
--E
+
+--S 41
+t1:=integrate(1/(p+q*sinh(a*x)),x)
+--R
+--R (2)
+--R log
+--R 2 2 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q - 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R /
+--R +-------+
+--R | 2 2
+--R a\|q + p
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 42
+bb:=x/q-p/q*t1
+--R
+--R (3)
+--R -
+--R p
+--R *
+--R log
+--R 2 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x)
+--R +
+--R 2 2 2 2
+--R q cosh(a x) + 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q
- 2p
+--R /
+--R 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R +-------+
+--R | 2 2
+--R a x\|q + p
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc:=aa-bb
+--R
+--R (4)
+--R p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R +
+--R p
+--R *
+--R log
+--R 2 2 2 2
2
+--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q
cosh(a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q -
2p
+--R /
+--R 2 2
+--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x)
+--R +
+--R 2p cosh(a x) - q
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 44
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (5) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 45
+dd:=sinhsqrrule cc
+--R
+--R (6)
+--R p
+--R *
+--R log
+--R 2 2
+--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x)
+--R +
+--R 2 2 2 2
+--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (4q + 4p q)sinh(a x) + (4q + 4p q)cosh(a x) + 4p q + 4p
+--R /
+--R 2
+--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x)
+--R +
+--R 4p cosh(a x) - 3q
+--R +
+--R p
+--R *
+--R log
+--R 2 2
+--R (4q cosh(a x) + 4p q)sinh(a x) + q cosh(2a x)
+--R +
+--R 2 2 2 2
+--R 2q cosh(a x) + 4p q cosh(a x) + q + 4p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 4q - 4p q)sinh(a x) + (- 4q - 4p q)cosh(a x) - 4p q -
4p
+--R /
+--R 2
+--R (4q cosh(a x) + 4p)sinh(a x) + q cosh(2a x) + 2q cosh(a x)
+--R +
+--R 4p cosh(a x) - 3q
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 46
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (7) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 47
+ee:=coshsqrrule dd
+--R
+--R (8)
+--R p
+--R *
+--R log
+--R 2 2
+--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2 3
+--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p
+--R /
+--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q
+--R +
+--R p
+--R *
+--R log
+--R 2 2
+--R (2q cosh(a x) + 2p q)sinh(a x) + q cosh(2a x)
+--R +
+--R 2 2
+--R 2p q cosh(a x) + q + 2p
+--R *
+--R +-------+
+--R | 2 2
+--R \|q + p
+--R +
+--R 3 2 3 2 2
3
+--R (- 2q - 2p q)sinh(a x) + (- 2q - 2p q)cosh(a x) - 2p q -
2p
+--R /
+--R (2q cosh(a x) + 2p)sinh(a x) + q cosh(2a x) + 2p cosh(a x) - q
+--R /
+--R +-------+
+--R | 2 2
+--R a q\|q + p
+--R Type: Expression
Integer
+--E
+
+--S 48 14:644 Schaums and Axiom differ by a constant
+ff:=complexNormalize ee
+--R
+--R 4 2 2
+--R p log(q + p q )
+--R (9) ----------------
+--R +-------+
+--R | 2 2
+--R a q\|q + p
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.645~~~~~$\displaystyle
@@ -268,7 +997,7 @@ $$
<<*>>=
)clear all
---S 10 of 10
+--S 49 14:645 Axiom cannot compute this integral
aa:=integrate(csch(a*x)^n,x)
--R
--R
diff --git a/src/input/schaum34.input.pamphlet
b/src/input/schaum34.input.pamphlet
index 1a5c359..83d7061 100644
--- a/src/input/schaum34.input.pamphlet
+++ b/src/input/schaum34.input.pamphlet
@@ -18,7 +18,7 @@ $$
)set message auto off
)clear all
---S 1 of 32
+--S 1
aa:=integrate(asinh(x/a),x)
--R
--R
@@ -33,6 +33,69 @@ aa:=integrate(asinh(x/a),x)
--R \|x + a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=x*asinh(x/a)-sqrt(x^2+a^2)
+--R
+--R +-------+
+--R | 2 2 x
+--R (2) - \|x + a + x asinh(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x + a + x x
+--R (3) x log(--------------) - x asinh(-)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 4
+asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1)))
+--R
+--R +------+
+--R | 2
+--R (4) asinh(x) == log(\|x + 1 + x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 5
+dd:=asinhlogrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x + a
+--R +-------+ a |------- + x
+--R | 2 2 | 2
+--R \|x + a + x \| a
+--R (5) x log(--------------) - x log(---------------)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 6
+ee:=expandLog dd
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R | 2 2 |x + a
+--R (6) x log(\|x + a + x) - x log(a |------- + x)
+--R | 2
+--R \| a
+--R Type: Expression
Integer
+--E
+
+--S 7 14:646 Schaums and Axiom agree
+ff:=rootSimp ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.647~~~~~$\displaystyle
@@ -44,7 +107,7 @@ $$
<<*>>=
)clear all
---S 2 of 32
+--S 8
aa:=integrate(x*asinh(x/a),x)
--R
--R
@@ -64,6 +127,77 @@ aa:=integrate(x*asinh(x/a),x)
--R 8x\|x + a - 8x - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 9
+bb:=(x^2/2+a^2/4)*asinh(x/a)-(x*sqrt(x^2+a^2))/4
+--R
+--R +-------+
+--R | 2 2 2 2 x
+--R - x\|x + a + (2x + a )asinh(-)
+--R a
+--R (2) ----------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 10
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R 2 2 \|x + a + x 2 2 x
+--R (2x + a )log(--------------) + (- 2x - a )asinh(-)
+--R a a
+--R (3) ----------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 11
+asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1)))
+--R
+--R +------+
+--R | 2
+--R (4) asinh(x) == log(\|x + 1 + x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 12
+dd:=asinhlogrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x + a
+--R +-------+ a |------- + x
+--R | 2 2 | 2
+--R 2 2 \|x + a + x 2 2 \| a
+--R (2x + a )log(--------------) + (- 2x - a )log(---------------)
+--R a a
+--R (5) ----------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 13
+ee:=expandLog dd
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R 2 2 | 2 2 2 2 |x + a
+--R (2x + a )log(\|x + a + x) + (- 2x - a )log(a |------- + x)
+--R | 2
+--R \| a
+--R (6) ----------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 14 14:647 Schaums and Axiom agree
+ff:=rootSimp ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.648~~~~~$\displaystyle
@@ -74,7 +208,7 @@ $$
<<*>>=
)clear all
---S 3 of 32
+--S 15
aa:=integrate(x^2*asinh(x/a),x)
--R
--R
@@ -94,6 +228,77 @@ aa:=integrate(x^2*asinh(x/a),x)
--R (36x + 9a )\|x + a - 36x - 27a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 16
+bb:=x^3/3*asinh(x/a)+((2*a^2-x^2)*sqrt(x^2+a^2))/9
+--R
+--R +-------+
+--R 2 2 | 2 2 3 x
+--R (- x + 2a )\|x + a + 3x asinh(-)
+--R a
+--R (2) ------------------------------------
+--R 9
+--R Type: Expression
Integer
+--E
+
+--S 17
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R 3 \|x + a + x 3 x
+--R x log(--------------) - x asinh(-)
+--R a a
+--R (3) ----------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 18
+asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1)))
+--R
+--R +------+
+--R | 2
+--R (4) asinh(x) == log(\|x + 1 + x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 19
+dd:=asinhlogrule cc
+--R
+--R +-------+
+--R | 2 2
+--R |x + a
+--R +-------+ a |------- + x
+--R | 2 2 | 2
+--R 3 \|x + a + x 3 \| a
+--R x log(--------------) - x log(---------------)
+--R a a
+--R (5) ----------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 20
+ee:=expandLog dd
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R 3 | 2 2 3 |x + a
+--R x log(\|x + a + x) - x log(a |------- + x)
+--R | 2
+--R \| a
+--R (6) ----------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 21 14:648 Schaums and Axiom agree
+ff:=rootSimp ee
+--R
+--R (7) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.649~~~~~$\displaystyle
@@ -124,7 +329,7 @@ $$
<<*>>=
)clear all
---S 4 of 32
+--S 22 14:649 Axiom cannot compute this integral
aa:=integrate(asinh(x/a)/x,x)
--R
--R
@@ -146,7 +351,7 @@ $$
<<*>>=
)clear all
---S 5 of 32
+--S 23
aa:=integrate(asinh(x/a)/x^2,x)
--R
--R
@@ -164,6 +369,113 @@ aa:=integrate(asinh(x/a)/x^2,x)
--R a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=-asinh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2
+--R \|x + a + a x
+--R - x log(--------------) - a asinh(-)
+--R x a
+--R (2) ------------------------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - x log(\|x + a - x + a) + x log(\|x + a - x - a)
+--R +
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R \|x + a + x \|x + a + a x
+--R - a log(--------------) + x log(--------------) + a asinh(-)
+--R a x a
+--R /
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 26
+asinhlogrule:=rule(asinh(x) == log(x+sqrt(x^2+1)))
+--R
+--R +------+
+--R | 2
+--R (4) asinh(x) == log(\|x + 1 + x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 27
+dd:=asinhlogrule cc
+--R
+--R (5)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - x log(\|x + a - x + a) + x log(\|x + a - x - a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x + a
+--R +-------+ +-------+ a |-------
+ x
+--R | 2 2 | 2 2 | 2
+--R \|x + a + x \|x + a + a \| a
+--R - a log(--------------) + x log(--------------) + a
log(---------------)
+--R a x a
+--R /
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 28
+ee:=expandLog dd
+--R
+--R (6)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - a log(\|x + a + x) + x log(\|x + a + a)
+--R +
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - x log(\|x + a - x + a) + x log(\|x + a - x - a)
+--R +
+--R +-------+
+--R | 2 2
+--R |x + a
+--R a log(a |------- + x) - x log(x)
+--R | 2
+--R \| a
+--R /
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 29
+ff:=rootSimp ee
+--R
+--R (7)
+--R +-------+ +-------+ +-------+
+--R | 2 2 | 2 2 | 2 2
+--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x -
a)
+--R +
+--R - log(x)
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 30 14:650 Schaums and Axiom differ by a constant
+gg:=complexNormalize ff
+--R
+--R log(- 1)
+--R (8) - --------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.651~~~~~$\displaystyle
@@ -182,7 +494,7 @@ $$
<<*>>=
)clear all
---S 6 of 32
+--S 31
aa:=integrate(acosh(x/a),x)
--R
--R
@@ -197,6 +509,100 @@ aa:=integrate(acosh(x/a),x)
--R \|x - a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb1:=x*acosh(x/a)-sqrt(x^2-a^2)
+--R
+--R +-------+
+--R | 2 2 x
+--R (2) - \|x - a + x acosh(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 33
+bb2:=x*acosh(x/a)+sqrt(x^2-a^2)
+--R
+--R +-------+
+--R | 2 2 x
+--R (3) \|x - a + x acosh(-)
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 34
+cc1:=aa-bb1
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a + x x
+--R (4) x log(--------------) - x acosh(-)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 35
+cc2:=aa-bb2
+--R
+--R (5)
+--R +-------+
+--R +-------+ | 2 2 +-------+
+--R | 2 2 2 \|x - a + x x | 2 2
+--R (x\|x - a - x )log(--------------) + (- x acosh(-) + 2x)\|x - a
+--R a a
+--R +
+--R 2 x 2 2
+--R x acosh(-) - 2x + 2a
+--R a
+--R /
+--R +-------+
+--R | 2 2
+--R \|x - a - x
+--R Type: Expression
Integer
+--E
+
+--S 36
+acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1)))
+--R
+--R +------+
+--R | 2
+--R (6) acosh(x) == log(\|x - 1 + x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 37
+dd1:=acoshlogrule cc1
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R +-------+ a |------- + x
+--R | 2 2 | 2
+--R \|x - a + x \| a
+--R (7) x log(--------------) - x log(---------------)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 38
+ee1:=expandLog dd1
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R | 2 2 |x - a
+--R (8) x log(\|x - a + x) - x log(a |------- + x)
+--R | 2
+--R \| a
+--R Type: Expression
Integer
+--E
+
+--S 39 14:651 Schaums and Axiom agree
+ff1:=rootSimp ee1
+--R
+--R (9) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.652~~~~~$\displaystyle
@@ -217,7 +623,7 @@ $$
<<*>>=
)clear all
---S 7 of 32
+--S 40
aa:=integrate(x*acosh(x/a),x)
--R
--R
@@ -237,6 +643,114 @@ aa:=integrate(x*acosh(x/a),x)
--R 8x\|x - a - 8x + 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 41
+bb1:=1/4*(2*x^2-a^2)*acosh(x/a)-1/4*x*sqrt(x^2-a^2)
+--R
+--R +-------+
+--R | 2 2 2 2 x
+--R - x\|x - a + (2x - a )acosh(-)
+--R a
+--R (2) ----------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 42
+bb2:=1/4*(2*x^2-a^2)*acosh(x/a)+1/4*x*sqrt(x^2-a^2)
+--R
+--R +-------+
+--R | 2 2 2 2 x
+--R x\|x - a + (2x - a )acosh(-)
+--R a
+--R (3) --------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc1:=aa-bb1
+--R
+--R +-------+
+--R | 2 2
+--R 2 2 \|x - a + x 2 2 x
+--R (2x - a )log(--------------) + (- 2x + a )acosh(-)
+--R a a
+--R (4) ----------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 44
+cc2:=aa-bb2
+--R
+--R (5)
+--R +-------+
+--R +-------+ | 2 2
+--R 3 2 | 2 2 4 2 2 4 \|x - a + x
+--R ((4x - 2a x)\|x - a - 4x + 4a x - a )log(--------------)
+--R a
+--R +
+--R +-------+
+--R 3 2 x 3 2 | 2 2
+--R ((- 4x + 2a x)acosh(-) + 4x - 2a x)\|x - a
+--R a
+--R +
+--R 4 2 2 4 x 4 2 2
+--R (4x - 4a x + a )acosh(-) - 4x + 4a x
+--R a
+--R /
+--R +-------+
+--R | 2 2 2 2
+--R 8x\|x - a - 8x + 4a
+--R Type: Expression
Integer
+--E
+
+--S 45
+acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1)))
+--R
+--R +------+
+--R | 2
+--R (6) acosh(x) == log(\|x - 1 + x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 46
+dd1:=acoshlogrule cc1
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R +-------+ a |------- + x
+--R | 2 2 | 2
+--R 2 2 \|x - a + x 2 2 \| a
+--R (2x - a )log(--------------) + (- 2x + a )log(---------------)
+--R a a
+--R (7) ----------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 47
+ee1:=expandLog dd1
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R 2 2 | 2 2 2 2 |x - a
+--R (2x - a )log(\|x - a + x) + (- 2x + a )log(a |------- + x)
+--R | 2
+--R \| a
+--R (8) ----------------------------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 48 14:652 Schaums and Axiom agree
+ff1:=rootSimp ee1
+--R
+--R (9) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.653~~~~~$\displaystyle
@@ -257,7 +771,7 @@ $$
<<*>>=
)clear all
---S 8 of 32
+--S 49
aa:=integrate(x^2*acosh(x/a),x)
--R
--R
@@ -277,6 +791,114 @@ aa:=integrate(x^2*acosh(x/a),x)
--R (36x - 9a )\|x - a - 36x + 27a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 50
+bb1:=1/3*x^3*acosh(x/a)-1/9*(x^2+2*a^2)*sqrt(x^2-a^2)
+--R
+--R +-------+
+--R 2 2 | 2 2 3 x
+--R (- x - 2a )\|x - a + 3x acosh(-)
+--R a
+--R (2) ------------------------------------
+--R 9
+--R Type: Expression
Integer
+--E
+
+--S 51
+bb2:=1/3*x^3*acosh(x/a)+1/9*(x^2+2*a^2)*sqrt(x^2-a^2)
+--R
+--R +-------+
+--R 2 2 | 2 2 3 x
+--R (x + 2a )\|x - a + 3x acosh(-)
+--R a
+--R (3) ----------------------------------
+--R 9
+--R Type: Expression
Integer
+--E
+
+--S 52
+cc1:=aa-bb1
+--R
+--R +-------+
+--R | 2 2
+--R 3 \|x - a + x 3 x
+--R x log(--------------) - x acosh(-)
+--R a a
+--R (4) ----------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 53
+cc2:=aa-bb2
+--R
+--R (5)
+--R +-------+
+--R +-------+ | 2 2
+--R 5 2 3 | 2 2 6 2 4 \|x - a + x
+--R ((12x - 3a x )\|x - a - 12x + 9a x )log(--------------)
+--R a
+--R +
+--R +-------+
+--R 5 2 3 x 5 2 3 4 | 2 2
+--R ((- 12x + 3a x )acosh(-) + 8x + 10a x - 12a x)\|x - a
+--R a
+--R +
+--R 6 2 4 x 6 2 4 4 2 6
+--R (12x - 9a x )acosh(-) - 8x - 6a x + 18a x - 4a
+--R a
+--R /
+--R +-------+
+--R 2 2 | 2 2 3 2
+--R (36x - 9a )\|x - a - 36x + 27a x
+--R Type: Expression
Integer
+--E
+
+--S 54
+acoshlogrule:=rule(acosh(x) == log(x+sqrt(x^2-1)))
+--R
+--R +------+
+--R | 2
+--R (6) acosh(x) == log(\|x - 1 + x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 55
+dd1:=acoshlogrule cc1
+--R
+--R +-------+
+--R | 2 2
+--R |x - a
+--R +-------+ a |------- + x
+--R | 2 2 | 2
+--R 3 \|x - a + x 3 \| a
+--R x log(--------------) - x log(---------------)
+--R a a
+--R (7) ----------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 56
+ee1:=expandLog dd1
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R 3 | 2 2 3 |x - a
+--R x log(\|x - a + x) - x log(a |------- + x)
+--R | 2
+--R \| a
+--R (8) ----------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 57 14:653 Schaums and Axiom agree
+ff1:=rootSimp ee1
+--R
+--R (9) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.654~~~~~$\displaystyle
@@ -296,7 +918,7 @@ $$
<<*>>=
)clear all
---S 9 of 32
+--S 58 14:654 Axiom cannot compute this integral
aa:=integrate(acosh(x/a)/x,x)
--R
--R
@@ -325,7 +947,7 @@ $$
<<*>>=
)clear all
---S 10 of 32
+--S 59
aa:=integrate(acosh(x/a)/x^2,x)
--R
--R
@@ -338,6 +960,68 @@ aa:=integrate(acosh(x/a)/x^2,x)
--R a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 60
+bb1:=-acosh(x/a)/x-1/a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2
+--R \|x + a + a x
+--R - x log(--------------) - a acosh(-)
+--R x a
+--R (2) ------------------------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 61
+bb2:=-acosh(x/a)/x+1/a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2
+--R \|x + a + a x
+--R x log(--------------) - a acosh(-)
+--R x a
+--R (3) ----------------------------------
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 62
+cc1:=aa-bb1
+--R
+--R (4)
+--R +-------+ +-------+ +-------+
+--R | 2 2 | 2 2 | 2 2
+--R \|x + a + a \|x - a + x \|x - a -
x
+--R x log(--------------) - a log(--------------) + 2x
atan(--------------)
+--R x a a
+--R +
+--R x
+--R a acosh(-)
+--R a
+--R /
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 63 14:655 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +-------+ +-------+ +-------+
+--R | 2 2 | 2 2 | 2 2
+--R \|x + a + a \|x - a + x \|x - a
- x
+--R - x log(--------------) - a log(--------------) + 2x
atan(--------------)
+--R x a a
+--R +
+--R x
+--R a acosh(-)
+--R a
+--R /
+--R a x
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.656~~~~~$\displaystyle
@@ -348,7 +1032,7 @@ $$
<<*>>=
)clear all
---S 11 of 32
+--S 64
aa:=integrate(atanh(x/a),x)
--R
--R
@@ -359,6 +1043,58 @@ aa:=integrate(atanh(x/a),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 65
+bb:=x*atanh(x/a)+a/2*log(a^2-x^2)
+--R
+--R 2 2 x
+--R a log(- x + a ) + 2x atanh(-)
+--R a
+--R (2) ------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 66
+cc:=aa-bb
+--R
+--R 2 2 - x - a 2 2 x
+--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-)
+--R x - a a
+--R (3) ----------------------------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 67
+atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x)))
+--R
+--R - x - 1
+--R log(-------)
+--R x - 1
+--R (4) atanh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 68
+dd:=atanhrule cc
+--R
+--R 2 2 2 2
+--R a log(x - a ) - a log(- x + a )
+--R (5) ---------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 69 14:656 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R a log(- 1)
+--R (6) ----------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.657~~~~~$\displaystyle
@@ -369,7 +1105,7 @@ $$
<<*>>=
)clear all
---S 12 of 32
+--S 70
aa:=integrate(x*atanh(x/a),x)
--R
--R
@@ -380,6 +1116,46 @@ aa:=integrate(x*atanh(x/a),x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 71
+bb:=(a*x)/2+1/2*(x^2-a^2)*atanh(x/a)
+--R
+--R 2 2 x
+--R (x - a )atanh(-) + a x
+--R a
+--R (2) -----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 72
+cc:=aa-bb
+--R
+--R 2 2 - x - a 2 2 x
+--R (x - a )log(-------) + (- 2x + 2a )atanh(-)
+--R x - a a
+--R (3) ---------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 73
+atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x)))
+--R
+--R - x - 1
+--R log(-------)
+--R x - 1
+--R (4) atanh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 74 14:657 Schaums and Axiom agree
+dd:=atanhrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.658~~~~~$\displaystyle
@@ -391,7 +1167,7 @@ $$
<<*>>=
)clear all
---S 13 of 32
+--S 75
aa:=integrate(x^2*atanh(x/a),x)
--R
--R
@@ -402,6 +1178,59 @@ aa:=integrate(x^2*atanh(x/a),x)
--R 6
--R Type: Union(Expression
Integer,...)
--E
+
+--S 76
+bb:=(a*x^2)/6+x^3/3*atanh(x/a)+a^3/6*log(a^2-x^2)
+--R
+--R 3 2 2 3 x 2
+--R a log(- x + a ) + 2x atanh(-) + a x
+--R a
+--R (2) -------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 77
+cc:=aa-bb
+--R
+--R 3 2 2 3 - x - a 3 2 2 3 x
+--R a log(x - a ) + x log(-------) - a log(- x + a ) - 2x atanh(-)
+--R x - a a
+--R (3) ----------------------------------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 78
+atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x)))
+--R
+--R - x - 1
+--R log(-------)
+--R x - 1
+--R (4) atanh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 79
+dd:=atanhrule cc
+--R
+--R 3 2 2 3 2 2
+--R a log(x - a ) - a log(- x + a )
+--R (5) ---------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 80 14:658 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R 3
+--R a log(- 1)
+--R (6) ----------
+--R 6
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.659~~~~~$\displaystyle
@@ -412,7 +1241,7 @@ $$
<<*>>=
)clear all
---S 14 of 32
+--S 81 14:659 Axiom cannot compute this integral
aa:=integrate(atanh(x/a)/x,x)
--R
--R
@@ -433,7 +1262,7 @@ $$
<<*>>=
)clear all
---S 15 of 32
+--S 82
aa:=integrate(atanh(x/a)/x^2,x)
--R
--R
@@ -444,6 +1273,70 @@ aa:=integrate(atanh(x/a)/x^2,x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 83
+bb:=-atanh(x/a)/x+1/(2*a)*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R x x
+--R x log(- -------) - 2a atanh(-)
+--R 2 2 a
+--R x - a
+--R (2) ------------------------------
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 84
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R 2 2 x - x - a
+--R - x log(x - a ) + 2x log(x) - x log(- -------) - a log(-------)
+--R 2 2 x - a
+--R x - a
+--R +
+--R x
+--R 2a atanh(-)
+--R a
+--R /
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 85
+atanhrule:=rule(atanh(x) == 1/2*log((1+x)/(1-x)))
+--R
+--R - x - 1
+--R log(-------)
+--R x - 1
+--R (4) atanh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 86
+dd:=atanhrule cc
+--R
+--R 2
+--R 2 2 x
+--R - log(x - a ) + 2log(x) - log(- -------)
+--R 2 2
+--R x - a
+--R (5) -----------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 87 14:660 Schaums and Axiom agree
+ee:=expandLog dd
+--R
+--R log(- 1)
+--R (6) - --------
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.661~~~~~$\displaystyle
@@ -451,10 +1344,15 @@ aa:=integrate(atanh(x/a)/x^2,x)
$$\int{\coth^{-1}\frac{x}{a}}=
x\coth^{-1}{x}+\frac{a}{2}\ln(x^2-a^2)
$$
+
+Note that it appears there is a typo in Schaums (1968 printing 4).
+$$\int{\coth^{-1}\frac{x}{a}}=
+x\coth^{-1}{x/a}+\frac{a}{2}\ln(x^2-a^2)
+$$
<<*>>=
)clear all
---S 16 of 32
+--S 88
aa:=integrate(acoth(x/a),x)
--R
--R
@@ -465,6 +1363,46 @@ aa:=integrate(acoth(x/a),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 89
+bb:=x*acoth(x/a)+a/2*log(x^2-a^2)
+--R
+--R 2 2 x
+--R a log(x - a ) + 2x acoth(-)
+--R a
+--R (2) ----------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 90
+cc:=aa-bb
+--R
+--R x + a x
+--R x log(-----) - 2x acoth(-)
+--R x - a a
+--R (3) --------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 91
+acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1)))
+--R
+--R x + 1
+--R log(-----)
+--R x - 1
+--R (4) acoth(x) == ----------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 92 14:661 Schaums and Axiom agree
+dd:=acothrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.662~~~~~$\displaystyle
@@ -475,7 +1413,7 @@ $$
<<*>>=
)clear all
---S 17 of 32
+--S 93
aa:=integrate(x*acoth(x/a),x)
--R
--R
@@ -486,6 +1424,46 @@ aa:=integrate(x*acoth(x/a),x)
--R 4
--R Type: Union(Expression
Integer,...)
--E
+
+--S 94
+bb:=(a*x)/2+1/2*(x^2-a^2)*acoth(x/a)
+--R
+--R 2 2 x
+--R (x - a )acoth(-) + a x
+--R a
+--R (2) -----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 95
+cc:=aa-bb
+--R
+--R 2 2 x + a 2 2 x
+--R (x - a )log(-----) + (- 2x + 2a )acoth(-)
+--R x - a a
+--R (3) -------------------------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+--S 96
+acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1)))
+--R
+--R x + 1
+--R log(-----)
+--R x - 1
+--R (4) acoth(x) == ----------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 97 14:662 Schaums and Axiom agree
+dd:=acothrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.663~~~~~$\displaystyle
@@ -497,7 +1475,7 @@ $$
<<*>>=
)clear all
---S 18 of 32
+--S 98
aa:=integrate(x^2*acoth(x/a),x)
--R
--R
@@ -508,6 +1486,46 @@ aa:=integrate(x^2*acoth(x/a),x)
--R 6
--R Type: Union(Expression
Integer,...)
--E
+
+--S 99
+bb:=(a*x^2)/6+x^3/3*acoth(x/a)+a^3/6*log(x^2-a^2)
+--R
+--R 3 2 2 3 x 2
+--R a log(x - a ) + 2x acoth(-) + a x
+--R a
+--R (2) -----------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 100
+cc:=aa-bb
+--R
+--R 3 x + a 3 x
+--R x log(-----) - 2x acoth(-)
+--R x - a a
+--R (3) --------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 101
+acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1)))
+--R
+--R x + 1
+--R log(-----)
+--R x - 1
+--R (4) acoth(x) == ----------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 102 14:663 Schaums and Axiom agree
+dd:=acothrule cc
+--R
+--R (5) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.664~~~~~$\displaystyle
@@ -518,7 +1536,7 @@ $$
<<*>>=
)clear all
---S 19 of 32
+--S 103 14:664 Axiom cannot compute this integral
aa:=integrate(acoth(x/a)/x,x)
--R
--R
@@ -539,7 +1557,7 @@ $$
<<*>>=
)clear all
---S 20 of 32
+--S 104
aa:=integrate(acoth(x/a)/x^2,x)
--R
--R
@@ -550,6 +1568,64 @@ aa:=integrate(acoth(x/a)/x^2,x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 105
+bb:=-acoth(x/a)/x+1/(2*a)*log(x^2/(x^2-a^2))
+--R
+--R 2
+--R x x
+--R x log(-------) - 2a acoth(-)
+--R 2 2 a
+--R x - a
+--R (2) ----------------------------
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 106
+cc:=aa-bb
+--R
+--R (3)
+--R 2
+--R 2 2 x + a x x
+--R - x log(x - a ) + 2x log(x) - a log(-----) - x log(-------) + 2a
acoth(-)
+--R x - a 2 2 a
+--R x - a
+--R
--------------------------------------------------------------------------
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 107
+acothrule:=rule(acoth(x) == 1/2*log((x+1)/(x-1)))
+--R
+--R x + 1
+--R log(-----)
+--R x - 1
+--R (4) acoth(x) == ----------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 108
+dd:=acothrule cc
+--R
+--R 2
+--R 2 2 x
+--R - log(x - a ) + 2log(x) - log(-------)
+--R 2 2
+--R x - a
+--R (5) ---------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 109 14:665 Schaums and Axiom agree
+ee:=expandLog dd
+--R
+--R (6) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.666~~~~~$\displaystyle
@@ -568,7 +1644,7 @@ $$
<<*>>=
)clear all
---S 21 of 32
+--S 110
aa:=integrate(asech(x/a),x)
--R
--R
@@ -579,7 +1655,308 @@ aa:=integrate(asech(x/a),x)
--R x x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 111
+bb1:=x*asech(x/a)+a*asin(x/a)
+--R
+--R x x
+--R (2) a asin(-) + x asech(-)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 112
+bb2:=x*asech(x/a)-a*asin(x/a)
+--R
+--R x x
+--R (3) - a asin(-) + x asech(-)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 113
+cc1:=aa-bb1
+--R
+--R (4)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a x
x
+--R x log(----------------) - 2a atan(----------------) - a asin(-) - x
asech(-)
+--R x x a
a
+--R Type: Expression
Integer
+--E
+
+--S 114
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a x
x
+--R x log(----------------) - 2a atan(----------------) + a asin(-) - x
asech(-)
+--R x x a
a
+--R Type: Expression
Integer
+--E
+
+--S 115
+asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1)))
+--R
+--R +--------+
+--R | 2
+--R |- x + 1
+--R x |-------- + 1
+--R | 2
+--R \| x
+--R (6) asech(x) == log(----------------)
+--R x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 116
+dd1:=asechrule cc1
+--R
+--R (7)
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R x |--------- + a +---------+
+--R | 2 | 2 2
+--R \| x \|- x + a + a
+--R - x log(-----------------) + x log(----------------)
+--R x x
+--R +
+--R +---------+
+--R | 2 2
+--R \|- x + a - a x
+--R - 2a atan(----------------) - a asin(-)
+--R x a
+--R Type: Expression
Integer
+--E
+
+--S 117
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R +--------+
+--R | 2
+--R (8) asin(x) == %i log(\|- x + 1 - %i x)
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 118
+ee1:=asinrule dd1
+--R
+--R (9)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R |- x + a |- x + a
+--R x |--------- + a a |--------- - %i x
+--R | 2 | 2
+--R \| x \| a
+--R - x log(-----------------) - %i a log(--------------------)
+--R x a
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R x log(----------------) - 2a atan(----------------)
+--R x x
+--R Type: Expression Complex
Integer
+--E
+
+--S 119
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R - x + %i
+--R %i log(--------)
+--R x + %i
+--R (10) atan(x) == - ----------------
+--R 2
+--R Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 120
+ff1:=atanrule ee1
+--R
+--R (11)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R |- x + a |- x + a
+--R x |--------- + a a |--------- - %i x
+--R | 2 | 2
+--R \| x \| a
+--R - x log(-----------------) - %i a log(--------------------)
+--R x a
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a - \|- x + a + %i x + a
+--R x log(----------------) + %i a log(-------------------------)
+--R x +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R Type: Expression Complex
Integer
+--E
+
+--S 121
+gg1:=expandLog ff1
+--R
+--R (12)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R |- x + a |- x + a
+--R - x log(x |--------- + a) - %i a log(a |--------- - %i x)
+--R | 2 | 2
+--R \| x \| a
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a)
+--R +
+--R +---------+
+--R | 2 2
+--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 122
+hh1:=rootSimp gg1
+--R
+--R (13)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R | 2 2
+--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 123 14:666 Schaums and Axiom agree
+ii1:=complexNormalize hh1
+--R
+--R (14) 0
+--R Type: Expression Complex
Integer
+--E
+
+@
+Note that Axiom has a built-in assumption about the sign of asech(x/a).
+We can see this if we simplify the cc2 value and show that it differs
+by a complex value of x.
+<<*>>=
+--S 124
+dd2:=asechrule cc2
+--R
+--R (15)
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R x |--------- + a +---------+
+--R | 2 | 2 2
+--R \| x \|- x + a + a
+--R - x log(-----------------) + x log(----------------)
+--R x x
+--R +
+--R +---------+
+--R | 2 2
+--R \|- x + a - a x
+--R - 2a atan(----------------) + a asin(-)
+--R x a
+--R Type: Expression
Integer
+--E
+
+--S 125
+ee2:=asinrule dd2
+--R
+--R (16)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R |- x + a |- x + a
+--R x |--------- + a a |--------- - %i x
+--R | 2 | 2
+--R \| x \| a
+--R - x log(-----------------) + %i a log(--------------------)
+--R x a
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a \|- x + a - a
+--R x log(----------------) - 2a atan(----------------)
+--R x x
+--R Type: Expression Complex
Integer
+--E
+
+--S 126
+ff2:=atanrule ee2
+--R
+--R (17)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R |- x + a |- x + a
+--R x |--------- + a a |--------- - %i x
+--R | 2 | 2
+--R \| x \| a
+--R - x log(-----------------) + %i a log(--------------------)
+--R x a
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R \|- x + a + a - \|- x + a + %i x + a
+--R x log(----------------) + %i a log(-------------------------)
+--R x +---------+
+--R | 2 2
+--R \|- x + a + %i x - a
+--R Type: Expression Complex
Integer
+--E
+
+--S 127
+gg2:=expandLog ff2
+--R
+--R (18)
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R |- x + a |- x + a
+--R - x log(x |--------- + a) + %i a log(a |--------- - %i x)
+--R | 2 | 2
+--R \| x \| a
+--R +
+--R +---------+ +---------+
+--R | 2 2 | 2 2
+--R - %i a log(\|- x + a + %i x - a) + x log(\|- x + a + a)
+--R +
+--R +---------+
+--R | 2 2
+--R %i a log(\|- x + a - %i x - a) - %i a log(a) + %i a log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 128
+hh2:=rootSimp gg2
+--R
+--R (19)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - %i a log(%i\|x - a + %i x - a) + %i a log(%i\|x - a - %i x)
+--R +
+--R +-------+
+--R | 2 2
+--R %i a log(%i\|x - a - %i x - a) - %i a log(a) + %i a log(- 1)
+--R Type: Expression Complex
Integer
+--E
+
+--S 129
+ii2:=complexNormalize hh2
+--R
+--R +-------+
+--R | 2 2
+--R (20) 2%i a log(%i\|x - a - %i x) - 2%i a log(a)
+--R Type: Expression Complex
Integer
+--E
+
@
+Thus we can conjecture that solutions that show up with x in only the
+imaginary part do so when the assumption of the sign of an inverse
+function differs.
\section{\cite{1}:14.667~~~~~$\displaystyle
\int{x{\rm ~sech}^{-1}\frac{x}{a}}~dx$}
@@ -599,7 +1976,7 @@ $$
<<*>>=
)clear all
---S 22 of 32
+--S 130
aa:=integrate(x*asech(x/a),x)
--R
--R
@@ -614,6 +1991,118 @@ aa:=integrate(x*asech(x/a),x)
--R 2\|- x + a - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 131
+bb1:=1/2*x^2*asech(x/a)-1/2*a*sqrt(a^2-x^2)
+--R
+--R +---------+
+--R | 2 2 2 x
+--R - a\|- x + a + x asech(-)
+--R a
+--R (2) ----------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 132
+bb2:=1/2*x^2*asech(x/a)+1/2*a*sqrt(a^2-x^2)
+--R
+--R +---------+
+--R | 2 2 2 x
+--R a\|- x + a + x asech(-)
+--R a
+--R (3) --------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 133
+cc1:=aa-bb1
+--R
+--R +---------+
+--R | 2 2
+--R 2 \|- x + a + a 2 x 2
+--R x log(----------------) - x asech(-) - a
+--R x a
+--R (4) -----------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 134
+cc2:=aa-bb2
+--R
+--R (5)
+--R +---------+
+--R +---------+ | 2 2
+--R 2 | 2 2 2 \|- x + a + a
+--R (x \|- x + a - a x )log(----------------)
+--R x
+--R +
+--R +---------+
+--R 2 x 2 | 2 2 2 x 2 3
+--R (- x asech(-) + a )\|- x + a + a x asech(-) + 2a x - a
+--R a a
+--R /
+--R +---------+
+--R | 2 2
+--R 2\|- x + a - 2a
+--R Type: Expression
Integer
+--E
+
+--S 135
+asechrule:=rule(asech(x) == log(1/x+sqrt(1/x^2-1)))
+--R
+--R +--------+
+--R | 2
+--R |- x + 1
+--R x |-------- + 1
+--R | 2
+--R \| x
+--R (6) asech(x) == log(----------------)
+--R x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 136
+dd1:=asechrule cc1
+--R
+--R +---------+
+--R | 2 2
+--R |- x + a
+--R x |--------- + a +---------+
+--R | 2 | 2 2
+--R 2 \| x 2 \|- x + a + a 2
+--R - x log(-----------------) + x log(----------------) - a
+--R x x
+--R (7) ---------------------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 137
+ee1:=expandLog dd1
+--R
+--R +---------+
+--R | 2 2 +---------+
+--R 2 |- x + a 2 | 2 2 2
+--R - x log(x |--------- + a) + x log(\|- x + a + a) - a
+--R | 2
+--R \| x
+--R (8) ---------------------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 138 14:667 Schaums and Axiom differ by a constant
+ff1:=rootSimp ee1
+--R
+--R 2
+--R a
+--R (9) - --
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.668~~~~~$\displaystyle
@@ -633,10 +2122,13 @@ $$\int{\frac{{\rm sech}^{-1}(x/a)}{x}}=
\end{array}
\right.
$$
+
+This is a interesting result since Axiom gives a closed form
+solution to the problem but Schaums gives a series solution.
<<*>>=
)clear all
---S 23 of 32
+--S 139 14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
aa:=integrate(asech(x/a)/x,x)
--R
--R
@@ -660,7 +2152,7 @@ $$
<<*>>=
)clear all
---S 24 of 32
+--S 140
aa:=integrate(acsch(x/a),x)
--R
--R
@@ -671,6 +2163,48 @@ aa:=integrate(acsch(x/a),x)
--R x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 141
+bb1:=x*acsch(x/a)+a*asinh(x/a)
+--R
+--R x x
+--R (2) a asinh(-) + x acsch(-)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 142
+bb2:=x*acsch(x/a)-a*asinh(x/a)
+--R
+--R x x
+--R (3) - a asinh(-) + x acsch(-)
+--R a a
+--R Type: Expression
Integer
+--E
+
+--S 143
+cc1:=aa-bb1
+--R
+--R (4)
+--R +-------+
+--R +-------+ | 2 2
+--R | 2 2 \|x + a + a x x
+--R - a log(\|x + a - x) + x log(--------------) - a asinh(-) - x acsch(-)
+--R x a a
+--R Type: Expression
Integer
+--E
+
+--S 144 14:669 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +-------+
+--R +-------+ | 2 2
+--R | 2 2 \|x + a + a x x
+--R - a log(\|x + a - x) + x log(--------------) + a asinh(-) - x acsch(-)
+--R x a a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.670~~~~~$\displaystyle
@@ -682,7 +2216,7 @@ $$
<<*>>=
)clear all
---S 25 of 32
+--S 145
aa:=integrate(x*acsch(x/a),x)
--R
--R
@@ -697,6 +2231,63 @@ aa:=integrate(x*acsch(x/a),x)
--R 2\|x + a - 2x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 146
+bb1:=x^2/2*acsch(x/a)+(a*sqrt(x^2+a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R a\|x + a + x acsch(-)
+--R a
+--R (2) ------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 147
+bb2:=x^2/2*acsch(x/a)-(a*sqrt(x^2+a^2))/2
+--R
+--R +-------+
+--R | 2 2 2 x
+--R - a\|x + a + x acsch(-)
+--R a
+--R (3) --------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 148
+cc1:=aa-bb1
+--R
+--R +-------+
+--R | 2 2
+--R 2 \|x + a + a 2 x
+--R x log(--------------) - x acsch(-)
+--R x a
+--R (4) ----------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 149 14:670 Axiom cannot simplify these expressions
+cc2:=aa-bb2
+--R
+--R (5)
+--R +-------+
+--R +-------+ | 2 2
+-------+
+--R 2 | 2 2 3 \|x + a + a 2 x | 2
2
+--R (x \|x + a - x )log(--------------) + (- x acsch(-) - 2a x)\|x +
a
+--R x a
+--R +
+--R 3 x 2 3
+--R x acsch(-) + 2a x + 2a
+--R a
+--R /
+--R +-------+
+--R | 2 2
+--R 2\|x + a - 2x
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.671~~~~~$\displaystyle
@@ -721,10 +2312,13 @@ $$\int{\frac{{\rm csch}^{-1}(x/a)}{x}}=
\end{array}
\right.
$$
+
+Schaums gives 3 different series expansions for this integral
+but Axiom has computed a closed form.
<<*>>=
)clear all
---S 26 of 32
+--S 150 14:671 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
aa:=integrate(acsch(x/a)/x,x)
--R
--R
@@ -737,6 +2331,7 @@ aa:=integrate(acsch(x/a)/x,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
@
\section{\cite{1}:14.672~~~~~$\displaystyle
@@ -748,7 +2343,7 @@ $$
<<*>>=
)clear all
---S 27 of 32
+--S 151 14:672 Axiom cannot compute this integral
aa:=integrate(x^m*asinh(x/a),x)
--R
--R
@@ -780,7 +2375,7 @@ $$
<<*>>=
)clear all
---S 28 of 32
+--S 152 14:673 Axiom cannot compute this integral
aa:=integrate(x^m*acosh(x/a),x)
--R
--R
@@ -801,7 +2396,7 @@ $$
<<*>>=
)clear all
---S 29 of 32
+--S 153 14:674 Axiom cannot compute this integral
aa:=integrate(x^m*atanh(x/a),x)
--R
--R
@@ -822,7 +2417,7 @@ $$
<<*>>=
)clear all
---S 30 of 32
+--S 154 14:675 Axiom cannot compute this integral
aa:=integrate(x^m*acoth(x/a),x)
--R
--R
@@ -854,7 +2449,7 @@ $$
<<*>>=
)clear all
---S 31 of 32
+--S 155 14:676 Axiom cannot compute this integral
aa:=integrate(x^m*asech(x/a),x)
--R
--R
@@ -877,7 +2472,7 @@ $$
<<*>>=
)clear all
---S 32 of 32
+--S 156 14:677 Axiom cannot compute this integral
aa:=integrate(x^m*acsch(x/a),x)
--R
--R
- [Axiom-developer] 20080427.01.tpd.patch (CATS Schaums-Axiom equivalence testing (29-34)),
daly <=