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[Axiom-developer] 20080417.02.tpd.patch (CATS Schaums-Axiom equivalence
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080417.02.tpd.patch (CATS Schaums-Axiom equivalence testing (8-10)) |
|
Date: |
Fri, 18 Apr 2008 01:54:04 -0500 |
schaum8.input.pamphlet
14:163 Schaums and Axiom differ by a constant
14:164 Schaums and Axiom differ by a constant
14:165 Schaums and Axiom differ by a constant
14:166 Schaums and Axiom differ by a constant
14:167 Schaums and Axiom differ by a constant
14:168 Schaums and Axiom differ by a constant
14:169 Schaums and Axiom differ by a constant
14:170 Schaums and Axiom differ by a constant
14:171 Schaums and Axiom agree
14:172 Schaums and Axiom differ by a constant
14:173 Schaums and Axiom differ by a constant
14:174 Schaums and Axiom differ by a constant
14:175 Schaums and Axiom differ by a constant
14:176 Schaums and Axiom differ by a constant
14:177 Axiom cannot do this integration
14:178 Axiom cannot simplify this expression
14:179 Axiom cannot integrate this expression
14:180 Axiom cannot integrate this expression
14:181 Axiom cannot integrate this expression
schaum9.input.pamphlet
14:182 Axiom cannot simplify this expression
14:183 Schaums and Axiom agree
14:184 Schaums and Axiom differ by a constant
14:185 Schaums and Axiom agree
14:186 Axiom cannot simplify this expression
14:187 Schaums and Axiom differ by a constant
14:188 Axiom cannot simplify this expression
14:189 Axiom cannot simplify this expression
14:190 Schaums and Axiom agree
14:191 Axiom cannot simplify this expression
14:192 Schaums and Axiom agree
14:193 Axiom cannot simplify this expression
14:194 Axiom cannot simplify this expression
14:195 Axiom cannot simplify this expression
14:196 Schaums and Axiom differ by a constant
14:197 Schaums and Axiom agree
14:198 Axiom cannot simplify this expression
14:199 Schaums and Axiom agree
14:200 Axiom cannot simplify this expression
14:201 Schaums and Axiom differ by a constant
14:202 Axiom cannot simplify this expression
14:203 Axiom cannot simplify this expression
14:204 Schaums and Axiom agree
14:205 Axiom cannot simplify this expression
14:206 Schaums and Axiom agree
14:207 Axiom cannot simplify this expression
14:208 Axiom cannot simplify this expression
14:209 Axiom cannot simplify this expression
schaum10.input.pamphlet
14:210 Schaums and Axiom differ by a constant
14:xxx Schaums and Axiom agree
14:211 Axiom cannot simplify this expression
14:212 Schaums and Axiom agree
14:213 Axiom cannot simplify this expression
14:214 Schaums and Axiom differ by a constant
14:215 Axiom cannot simplify this expression
14:216 Axiom cannot simplify this expression
14:217 Schaums and Axiom agree
14:218 Axiom cannot simplify this expression
14:219 Schaums and Axiom agree
14:220 Axiom cannot simplify this expression
14:221 Axiom cannot simplify this expression
14:222 Axiom cannot simplify this expression
14:223 Schaums and Axiom differ by a constant
14:224 Schaums and Axiom agree
14:225 Axiom cannot simplify this expression
14:226 Schaums and Axiom agree
14:227 Axiom cannot simplify this expression
14:228 Schaums and Axiom differ by a constant
14:229 Axiom cannot simplify this expression
14:230 Axiom cannot simplify this expression
14:231 Schaums and Axiom agree
14:232 Axiom cannot simplify this expression
14:233 Schaums and Axiom agree
14:234 Axiom cannot simplify this expression
14:235 Axiom cannot simplify this expression
14:236 Axiom cannot simplify this expression
========================================================================
diff --git a/changelog b/changelog
index 456e2f9..502aa1e 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence
+20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence
+20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum7.input fix typo in --S 30
20080417 tpd src/input/schaum5.input fix typo 14:88 -> 14:123
20080416 tpd src/input/schaum7.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum10.input.pamphlet
b/src/input/schaum10.input.pamphlet
index 8582086..d0ba7d9 100644
--- a/src/input/schaum10.input.pamphlet
+++ b/src/input/schaum10.input.pamphlet
@@ -15,7 +15,7 @@
$$\int{\frac{1}{\sqrt{x^2-a^2}}}=\ln\left(x+\sqrt{x^2-a^2}\right)$$
)set message auto off
)clear all
---S 1 of 28
+--S 1
aa:=integrate(1/(sqrt(x^2-a^2)),x)
--R
--R
@@ -24,6 +24,39 @@ aa:=integrate(1/(sqrt(x^2-a^2)),x)
--R (1) - log(\|x - a - x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=log(x+sqrt(x^2-a^2))
+--R
+--R +-------+
+--R | 2 2
+--R (2) log(\|x - a + x)
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x - a + x) - log(\|x - a - x)
+--R Type: Expression
Integer
+--E
+
+--S 4
+logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b))
+--R
+--I (4) c log(b) + c log(a) + %I == c log(a b) + %I
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 5 14:210 Schaums and Axiom differ by a constant
+dd:=logmul1 cc
+--R
+--R 2
+--R (5) - log(- a )
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.xxx~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2-a^2}}}$}
@@ -31,7 +64,7 @@ $$\int{\frac{x}{\sqrt{x^2-a^2}}}=\sqrt{x^2-a^2}$$
<<*>>=
)clear all
---S 2 of 28
+--S 6
aa:=integrate(x/(sqrt(x^2-a^2)),x)
--R
--R
@@ -44,6 +77,22 @@ aa:=integrate(x/(sqrt(x^2-a^2)),x)
--R \|x - a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 7
+bb:=sqrt(x^2-a^2)
+--R
+--R +-------+
+--R | 2 2
+--R (2) \|x - a
+--R Type: Expression
Integer
+--E
+
+--S 8 14:xxx Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.211~~~~~$\displaystyle
@@ -54,7 +103,7 @@ $$
<<*>>=
)clear all
---S 3 of 28
+--S 9
aa:=integrate(x^2/sqrt(x^2-a^2),x)
--R
--R
@@ -72,6 +121,29 @@ aa:=integrate(x^2/sqrt(x^2-a^2),x)
--R 4x\|x - a - 4x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 10
+bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 | 2 2
+--R a log(\|x - a + x) + x\|x - a
+--R (2) -----------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 11 14:211 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R - a log(\|x - a + x) - a log(\|x - a - x)
+--R (3) -----------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.212~~~~~$\displaystyle
@@ -82,7 +154,7 @@ $$
<<*>>=
)clear all
---S 4 of 28
+--S 12
aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R
--R
@@ -95,6 +167,24 @@ aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R (12x - 3a )\|x - a - 12x + 9a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 13
+bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2)
+--R
+--R +-------+
+--R 2 2 | 2 2
+--R (x + 2a )\|x - a
+--R (2) --------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 14 14:212 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.213~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2-a^2}}}$}
@@ -104,7 +194,7 @@ $$
<<*>>=
)clear all
---S 5 of 28
+--S 15
aa:=integrate(1/(x*sqrt(x^2-a^2)),x)
--R
--R
@@ -117,6 +207,30 @@ aa:=integrate(1/(x*sqrt(x^2-a^2)),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 16
+bb:=1/a*asec(abs(x/a))
+--R
+--R x
+--R asec(abs(-))
+--R a
+--R (2) ------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 17 14:213 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R 2atan(--------------) - asec(abs(-))
+--R a a
+--R (3) ------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.214~~~~~$\displaystyle
@@ -127,7 +241,7 @@ $$
<<*>>=
)clear all
---S 6 of 28
+--S 18
aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R
--R
@@ -138,6 +252,28 @@ aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R x\|x - a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 19
+bb:=sqrt(x^2-a^2)/(a^2*x)
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R (2) ----------
+--R 2
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 20 14:214 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 1
+--R (3) --
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.215~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2-a^2}}}$}
@@ -148,7 +284,7 @@ $$
<<*>>=
)clear all
---S 7 of 28
+--S 21
aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x)
--R
--R
@@ -168,6 +304,34 @@ aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x)
--R 4a x \|x - a - 4a x + 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 22
+bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 x | 2 2
+--R x asec(abs(-)) + a\|x - a
+--R a
+--R (2) ----------------------------
+--R 3 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 23 14:215 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R 2atan(--------------) - asec(abs(-))
+--R a a
+--R (3) ------------------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$}
@@ -177,7 +341,7 @@ $$
<<*>>=
)clear all
---S 8 of 28
+--S 24
aa:=integrate(sqrt(x^2-a^2),x)
--R
--R
@@ -195,6 +359,29 @@ aa:=integrate(sqrt(x^2-a^2),x)
--R 4x\|x - a - 4x + 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 25
+bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 | 2 2
+--R - a log(\|x - a + x) + x\|x - a
+--R (2) -------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 26 14:216 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R a log(\|x - a + x) + a log(\|x - a - x)
+--R (3) ---------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$}
@@ -204,7 +391,7 @@ $$
<<*>>=
)clear all
---S 9 of 28
+--S 27
aa:=integrate(x*sqrt(x^2-a^2),x)
--R
--R
@@ -217,6 +404,24 @@ aa:=integrate(x*sqrt(x^2-a^2),x)
--R (12x - 3a )\|x - a - 12x + 9a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 28
+bb:=(x^2-a^2)^(3/2)/3
+--R
+--R +-------+
+--R 2 2 | 2 2
+--R (x - a )\|x - a
+--R (2) -------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 29 14:217 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.218~~~~~$\displaystyle
@@ -228,7 +433,7 @@ $$
<<*>>=
)clear all
---S 10 of 28
+--S 30
aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R
--R
@@ -246,6 +451,28 @@ aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R (64x - 32a x)\|x - a - 64x + 64a x - 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 31
+bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 3 2 | 2 2
+--R - a log(\|x - a + x) + (2x - a x)\|x - a
+--R (2) -----------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 32 14:218 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R a log(\|x - a + x) + a log(\|x - a - x)
+--R (3) ---------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.219~~~~~$\displaystyle
@@ -256,7 +483,7 @@ $$
<<*>>=
)clear all
---S 11 of 28
+--S 33
aa:=integrate(x^3*sqrt(x^2-a^2),x)
--R
--R
@@ -273,6 +500,24 @@ aa:=integrate(x^3*sqrt(x^2-a^2),x)
--R (240x - 180a x + 15a )\|x - a - 240x + 300a x - 75a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 34
+bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3
+--R
+--R +-------+
+--R 4 2 2 4 | 2 2
+--R (3x - a x - 2a )\|x - a
+--R (2) ----------------------------
+--R 15
+--R Type: Expression
Integer
+--E
+
+--S 35 14:219 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.220~~~~~$\displaystyle
@@ -283,7 +528,7 @@ $$
<<*>>=
)clear all
---S 12 of 28
+--S 36
aa:=integrate(sqrt(x^2-a^2)/x,x)
--R
--R
@@ -298,6 +543,27 @@ aa:=integrate(sqrt(x^2-a^2)/x,x)
--R \|x - a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 37
+bb:=sqrt(x^2-a^2)-a*asec(abs(x/a))
+--R
+--R +-------+
+--R x | 2 2
+--R (2) - a asec(abs(-)) + \|x - a
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 38 14:220 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R (3) - 2a atan(--------------) + a asec(abs(-))
+--R a a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.221~~~~~$\displaystyle
@@ -308,7 +574,7 @@ $$
<<*>>=
)clear all
---S 13 of 28
+--S 39
aa:=integrate(sqrt(x^2-a^2)/x^2,x)
--R
--R
@@ -321,6 +587,27 @@ aa:=integrate(sqrt(x^2-a^2)/x^2,x)
--R x\|x - a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 40
+bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R x log(\|x - a + x) - \|x - a
+--R (2) ----------------------------------
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 41 14:221 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.222~~~~~$\displaystyle
@@ -332,7 +619,7 @@ $$
<<*>>=
)clear all
---S 14 of 28
+--S 42
aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R
--R
@@ -352,6 +639,32 @@ aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R 4a x \|x - a - 4a x + 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 43
+bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 x | 2 2
+--R x asec(abs(-)) - a\|x - a
+--R a
+--R (2) ----------------------------
+--R 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 44 14:222 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R 2atan(--------------) - asec(abs(-))
+--R a a
+--R (3) ------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$}
@@ -361,7 +674,7 @@ $$
<<*>>=
)clear all
---S 15 of 28
+--S 45
aa:=integrate(1/(x^2-a^2)^(3/2),x)
--R
--R
@@ -372,6 +685,27 @@ aa:=integrate(1/(x^2-a^2)^(3/2),x)
--R x\|x - a - x + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 46
+bb:=-x/(a^2*sqrt(x^2-a^2))
+--R
+--R x
+--R (2) - ------------
+--R +-------+
+--R 2 | 2 2
+--R a \|x - a
+--R Type: Expression
Integer
+--E
+
+--S 47 14:223 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 1
+--R (3) - --
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.224~~~~~$\displaystyle
@@ -382,7 +716,7 @@ $$
<<*>>=
)clear all
---S 16 of 28
+--S 48
aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R
--R
@@ -395,6 +729,24 @@ aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R x\|x - a - x + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 49
+bb:=-1/sqrt(x^2-a^2)
+--R
+--R 1
+--R (2) - ----------
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R Type: Expression
Integer
+--E
+
+--S 50 14:224 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.225~~~~~$\displaystyle
@@ -405,7 +757,7 @@ $$
<<*>>=
)clear all
---S 17 of 28
+--S 51
aa:=integrate(x^2/(x^2-a^2)^(3/2),x)
--R
--R
@@ -418,6 +770,28 @@ aa:=integrate(x^2/(x^2-a^2)^(3/2),x)
--R x\|x - a - x + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 52
+bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R \|x - a log(\|x - a + x) - x
+--R (2) ---------------------------------
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R Type: Expression
Integer
+--E
+
+--S 53 14:225 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.226~~~~~$\displaystyle
@@ -428,7 +802,7 @@ $$
<<*>>=
)clear all
---S 18 of 28
+--S 54
aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R
--R
@@ -441,6 +815,25 @@ aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R (2x - a )\|x - a - 2x + 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2)
+--R
+--R 2 2
+--R x - 2a
+--R (2) ----------
+--R +-------+
+--R | 2 2
+--R \|x - a
+--R Type: Expression
Integer
+--E
+
+--S 56 14:226 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.227~~~~~$\displaystyle
@@ -452,7 +845,7 @@ $$
<<*>>=
)clear all
---S 19 of 28
+--S 57
aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -467,6 +860,34 @@ aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x)
--R a x\|x - a - a x + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 58
+bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(abs(x/a))
+--R
+--R +-------+
+--R | 2 2 x
+--R - \|x - a asec(abs(-)) - a
+--R a
+--R (2) ----------------------------
+--R +-------+
+--R 3 | 2 2
+--R a \|x - a
+--R Type: Expression
Integer
+--E
+
+--S 59 14:227 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R - 2atan(--------------) + asec(abs(-))
+--R a a
+--R (3) --------------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.228~~~~~$\displaystyle
@@ -477,7 +898,7 @@ $$
<<*>>=
)clear all
---S 20 of 28
+--S 60
aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -488,6 +909,29 @@ aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R (2x - a x)\|x - a - 2x + 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 61
+bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2))
+--R
+--R 2 2
+--R - 2x + a
+--R (2) -------------
+--R +-------+
+--R 4 | 2 2
+--R a x\|x - a
+--R Type: Expression
Integer
+--E
+
+--S 62 14:228 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 2
+--R (3) - --
+--R 4
+--R a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.229~~~~~$\displaystyle
@@ -500,7 +944,7 @@ $$
<<*>>=
)clear all
---S 21 of 28
+--S 63
aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -520,6 +964,34 @@ aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x)
--R (8a x - 6a x )\|x - a - 8a x + 10a x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 64
+bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 | 2 2 x 2 3
+--R - 3x \|x - a asec(abs(-)) - 3a x + a
+--R a
+--R (2) ----------------------------------------
+--R +-------+
+--R 5 2 | 2 2
+--R 2a x \|x - a
+--R Type: Expression
Integer
+--E
+
+--S 65 14:229 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R - 6atan(--------------) + 3asec(abs(-))
+--R a a
+--R (3) ---------------------------------------
+--R 5
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$}
@@ -530,7 +1002,7 @@ $$
<<*>>=
)clear all
---S 22 of 28
+--S 66
aa:=integrate((x^2-a^2)^(3/2),x)
--R
--R
@@ -551,6 +1023,29 @@ aa:=integrate((x^2-a^2)^(3/2),x)
--R (64x - 32a x)\|x - a - 64x + 64a x - 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 67
+bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 3 2 | 2 2
+--R 3a log(\|x - a + x) + (2x - 5a x)\|x - a
+--R (2) -----------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 68 14:230 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R - 3a log(\|x - a + x) - 3a log(\|x - a - x)
+--R (3) -------------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$}
@@ -558,7 +1053,7 @@ $$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$
<<*>>=
)clear all
---S 23 of 28
+--S 69
aa:=integrate(x*(x^2-a^2)^(3/2),x)
--R
--R
@@ -575,6 +1070,24 @@ aa:=integrate(x*(x^2-a^2)^(3/2),x)
--R (80x - 60a x + 5a )\|x - a - 80x + 100a x - 25a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 70
+bb:=(x^2-a^2)^(5/2)/5
+--R
+--R +-------+
+--R 4 2 2 4 | 2 2
+--R (x - 2a x + a )\|x - a
+--R (2) ---------------------------
+--R 5
+--R Type: Expression
Integer
+--E
+
+--S 71 14:231 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.232~~~~~$\displaystyle\int{x^2(x^2-a^2)^{3/2}}~dx$}
@@ -586,7 +1099,7 @@ $$
<<*>>=
)clear all
---S 24 of 28
+--S 72
aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R
--R
@@ -614,6 +1127,28 @@ aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R (1536x - 1536a x + 288a x)\|x - a - 1536x + 2304a x - 864a x +
48a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 73
+bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 6 | 2 2 5 2 3 4 | 2 2
+--R 3a log(\|x - a + x) + (8x - 14a x + 3a x)\|x - a
+--R (2) --------------------------------------------------------
+--R 48
+--R Type: Expression
Integer
+--E
+
+--S 74 14:232 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 6 | 2 2 6 | 2 2
+--R - a log(\|x - a + x) - a log(\|x - a - x)
+--R (3) -----------------------------------------------
+--R 16
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$}
@@ -623,7 +1158,7 @@ $$
<<*>>=
)clear all
---S 25 of 28
+--S 75
aa:=integrate(x^3*(x^2-a^2)^(3/2),x)
--R
--R
@@ -652,6 +1187,24 @@ aa:=integrate(x^3*(x^2-a^2)^(3/2),x)
--R - 1960a x + 245a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 76
+bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5
+--R
+--R +-------+
+--R 6 2 4 4 2 6 | 2 2
+--R (5x - 8a x + a x + 2a )\|x - a
+--R (2) ------------------------------------
+--R 35
+--R Type: Expression
Integer
+--E
+
+--S 77 14:233 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.234~~~~~$\displaystyle
@@ -663,7 +1216,7 @@ $$
<<*>>=
)clear all
---S 26 of 28
+--S 78
aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R
--R
@@ -683,6 +1236,29 @@ aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R (12x - 3a )\|x - a - 12x + 9a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 79
+bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(abs(x/a))
+--R
+--R +-------+
+--R 3 x 2 2 | 2 2
+--R 3a asec(abs(-)) + (x - 4a )\|x - a
+--R a
+--R (2) --------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 80 14:234 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R 3 \|x - a - x 3 x
+--R (3) 2a atan(--------------) - a asec(abs(-))
+--R a a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.235~~~~~$\displaystyle
@@ -694,7 +1270,7 @@ $$
<<*>>=
)clear all
---S 27 of 28
+--S 81
aa:=integrate((x^2-a^2)^{3/2}/x^2,x)
--R
--R
@@ -712,6 +1288,28 @@ aa:=integrate((x^2-a^2)^{3/2}/x^2,x)
--R (8x - 2a x)\|x - a - 8x + 6a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 82
+bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 2 | 2 2
+--R - 3a x log(\|x - a + x) + (x + 2a )\|x - a
+--R (2) -------------------------------------------------
+--R 2x
+--R Type: Expression
Integer
+--E
+
+--S 83 14:235 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2 2
+--R 3a log(\|x - a + x) + 3a log(\|x - a - x) + 2a
+--R (3) -----------------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.236~~~~~$\displaystyle
@@ -723,7 +1321,7 @@ $$
<<*>>=
)clear all
---S 28 of 28
+--S 84
aa:=integrate((x^2-a^2)^(3/2)/x^3,x)
--R
--R
@@ -744,6 +1342,31 @@ aa:=integrate((x^2-a^2)^(3/2)/x^3,x)
--R Type: Union(Expression
Integer,...)
--E
+--S 85
+bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(abs(x/a))
+--R
+--R +-------+
+--R 2 x 2 2 | 2 2
+--R - 3a x asec(abs(-)) + (2x + a )\|x - a
+--R a
+--R (2) ------------------------------------------
+--R 2
+--R 2x
+--R Type: Expression
Integer
+--E
+
+--S 86 14:236 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+
+--R | 2 2
+--R \|x - a - x x
+--R - 6a atan(--------------) + 3a asec(abs(-))
+--R a a
+--R (3) -------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
)spool
)lisp (bye)
@
diff --git a/src/input/schaum8.input.pamphlet b/src/input/schaum8.input.pamphlet
index a5c0c7e..778d135 100644
--- a/src/input/schaum8.input.pamphlet
+++ b/src/input/schaum8.input.pamphlet
@@ -16,7 +16,7 @@ $$\int{\frac{1}{a^2-x^2}}=-\frac{1}{a}\coth^{-1}\frac{x}{a}$$
)set message auto off
)clear all
---S 1 of 19
+--S 1
aa:=integrate(1/(a^2-x^2),x)
--R
--R
@@ -25,6 +25,63 @@ aa:=integrate(1/(a^2-x^2),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=1/(2*a)*log((a+x)/(a-x))
+--R
+--R - x - a
+--R log(-------)
+--R x - a
+--R (2) ------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 3
+cc:=aa-bb
+--R
+--R - x - a
+--R log(x + a) - log(x - a) - log(-------)
+--R x - a
+--R (3) --------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 4
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 5
+dd:=divlog cc
+--R
+--R log(x + a) - log(- x - a)
+--R (5) -------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 6
+logminus:=rule(log(x + a) - log(- x - a) == log(-1))
+--R
+--I (6) log(x + a) - log(- x - a) + %I == log(- 1) + %I
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 7 14:163 Schaums and Axiom differ by a constant
+ee:=logminus dd
+--R
+--R log(- 1)
+--R (7) --------
+--R 2a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.164~~~~~$\displaystyle\int{\frac{x~dx}{a^2-x^2}}$}
@@ -32,7 +89,7 @@ $$\int{\frac{x}{a^2-x^2}}=-\frac{1}{2}\ln(a^2-x^2)$$
<<*>>=
)clear all
---S 2 of 19
+--S 8
aa:=integrate(x/(a^2-x^2),x)
--R
--R
@@ -42,6 +99,43 @@ aa:=integrate(x/(a^2-x^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 9
+bb:=-1/2*log(a^2-x^2)
+--R
+--R 2 2
+--R log(- x + a )
+--R (2) - --------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 10
+cc:=aa-bb
+--R
+--R 2 2 2 2
+--R - log(x - a ) + log(- x + a )
+--R (3) -------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 11
+logminus1:=rule(-log(x^2-a^2)+log(-x^2+a^2) == log(-1))
+--R
+--R 2 2 2 2
+--I (4) - log(x - a ) + log(- x + a ) + %H == log(- 1) + %H
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 12 14:164 Schaums and Axiom differ by a constant
+dd:=logminus1 cc
+--R
+--R log(- 1)
+--R (5) --------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.165~~~~~$\displaystyle\int{\frac{x^2~dx}{a^2-x^2}}$}
@@ -49,7 +143,7 @@
$$\int{\frac{x^2}{a^2-x^2}}=-x+\frac{a}{2}\ln\left(\frac{a+x}{a-x}\right)$$
<<*>>=
)clear all
---S 3 of 19
+--S 13
aa:=integrate(x^2/(a^2-x^2),x)
--R
--R
@@ -58,6 +152,62 @@ aa:=integrate(x^2/(a^2-x^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 14
+bb:=-x+a/2*log((a+x)/(a-x))
+--R
+--R - x - a
+--R a log(-------) - 2x
+--R x - a
+--R (2) -------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 15
+cc:=aa-bb
+--R
+--R - x - a
+--R a log(x + a) - a log(x - a) - a log(-------)
+--R x - a
+--R (3) --------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 16
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 17
+dd:=divlog cc
+--R
+--R a log(x + a) - a log(- x - a)
+--R (5) -----------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 18
+logminusa:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1))
+--R
+--I (6) b log(x + a) - b log(- x - a) + %M == b log(- 1) + %M
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 19 14:165 Schaums and Axiom differ by a constant
+ee:=logminusa dd
+--R
+--R a log(- 1)
+--R (7) ----------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.166~~~~~$\displaystyle\int{\frac{x^3~dx}{a^2-x^2}}$}
@@ -66,7 +216,7 @@
$$\int{\frac{x^3}{a^2-x^2}}=-\frac{x^2}{2}-\frac{a^2}{2}\ln(a^2-x^2)$$
<<*>>=
)clear all
---S 4 of 19
+--S 20
aa:=integrate(x^3/(a^2-x^2),x)
--R
--R
@@ -76,6 +226,45 @@ aa:=integrate(x^3/(a^2-x^2),x)
--R 2
--R Type: Union(Expression
Integer,...)
--E
+
+--S 21
+bb:=-x^2/2-a^2/2*log(a^2-x^2)
+--R
+--R 2 2 2 2
+--R - a log(- x + a ) - x
+--R (2) -----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 22
+cc:=aa-bb
+--R
+--R 2 2 2 2 2 2
+--R - a log(x - a ) + a log(- x + a )
+--R (3) -----------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 23
+logminus1b:=rule(-b*log(x^2-a^2)+b*log(-x^2+a^2) == b*log(-1))
+--R
+--R 2 2 2 2
+--I (4) - b log(x - a ) + b log(- x + a ) + %N == b log(- 1) + %N
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 24 14:166 Schaums and Axiom differ by a constant
+dd:=logminus1b cc
+--R
+--R 2
+--R a log(- 1)
+--R (5) ----------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.167~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)}}$}
@@ -85,7 +274,7 @@ $$
<<*>>=
)clear all
---S 5 of 19
+--S 25
aa:=integrate(1/(x*(a^2-x^2)),x)
--R
--R
@@ -96,6 +285,72 @@ aa:=integrate(1/(x*(a^2-x^2)),x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 26
+bb:=1/(2*a^2)*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R x
+--R log(- -------)
+--R 2 2
+--R x - a
+--R (2) --------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 27
+cc:=aa-bb
+--R
+--R 2
+--R 2 2 x
+--R - log(x - a ) + 2log(x) - log(- -------)
+--R 2 2
+--R x - a
+--R (3) -----------------------------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 28
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 29
+dd:=divlog cc
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 30
+logpowminus:=rule(log(-a^n) == n*log(a)+log(-1))
+--R
+--R n
+--R (6) log(- a ) == n log(a) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 31 14:167 Schaums and Axiom differ by a constant
+ee:=logpowminus dd
+--R
+--R log(- 1)
+--R (7) - --------
+--R 2
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.168~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)}}$}
@@ -105,7 +360,7 @@ $$
<<*>>=
)clear all
---S 6 of 19
+--S 32
aa:=integrate(1/(x^2*(a^2-x^2)),x)
--R
--R
@@ -115,6 +370,67 @@ aa:=integrate(1/(x^2*(a^2-x^2)),x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 33
+bb:=-1/(a^2*x)+1/(2*a^3)*log((a+x)/(a-x))
+--R
+--R - x - a
+--R x log(-------) - 2a
+--R x - a
+--R (2) -------------------
+--R 3
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 34
+cc:=aa-bb
+--R
+--R - x - a
+--R log(x + a) - log(x - a) - log(-------)
+--R x - a
+--R (3) --------------------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 35
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 36
+dd:=divlog cc
+--R
+--R log(x + a) - log(- x - a)
+--R (5) -------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 37
+logminus:=rule(log(x + a) - log(- x - a) == log(-1))
+--R
+--I (6) log(x + a) - log(- x - a) + %O == log(- 1) + %O
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 38 14:168 Schaums and Axiom differ by a constant
+ee:=logminus dd
+--R
+--R log(- 1)
+--R (7) --------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.169~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)}}$}
@@ -124,7 +440,7 @@ $$
<<*>>=
)clear all
---S 7 of 19
+--S 39
aa:=integrate(1/(x^3*(a^2-x^2)),x)
--R
--R
@@ -135,6 +451,70 @@ aa:=integrate(1/(x^3*(a^2-x^2)),x)
--R 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 40
+bb:=-1/(2*a^2*x^2)+1/(2*a^4)*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R 2 x 2
+--R x log(- -------) - a
+--R 2 2
+--R x - a
+--R (2) ---------------------
+--R 4 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 41
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 42
+bb1:=divlog bb
+--R
+--R 2 2 2 2 2 2
+--R - x log(x - a ) + x log(- x ) - a
+--R (4) -----------------------------------
+--R 4 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 43
+cc:=aa-bb1
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 44
+logminuspow:=rule(log(-x^n) == n*log(x)+log(-1))
+--R
+--R n
+--R (6) log(- x ) == n log(x) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 45 14:169 Schaums and Axiom differ by a constant
+dd:=logminuspow cc
+--R
+--R log(- 1)
+--R (7) - --------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.170~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^2}}$}
@@ -144,7 +524,7 @@ $$
<<*>>=
)clear all
---S 8 of 19
+--S 46
aa:=integrate(1/((a^2-x^2)^2),x)
--R
--R
@@ -155,6 +535,66 @@ aa:=integrate(1/((a^2-x^2)^2),x)
--R 4a x - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 47
+bb:=x/(2*a^2*(a^2-x^2))+1/(4*a^3)*log((a+x)/(a-x))
+--R
+--R 2 2 - x - a
+--R (x - a )log(-------) - 2a x
+--R x - a
+--R (2) ----------------------------
+--R 3 2 5
+--R 4a x - 4a
+--R Type: Expression
Integer
+--E
+
+--S 48
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 49
+bb1:=divlog bb
+--R
+--R 2 2 2 2
+--R (- x + a )log(x - a) + (x - a )log(- x - a) - 2a x
+--R (4) ----------------------------------------------------
+--R 3 2 5
+--R 4a x - 4a
+--R Type: Expression
Integer
+--E
+
+--S 50
+cc:=aa-bb1
+--R
+--R log(x + a) - log(- x - a)
+--R (5) -------------------------
+--R 3
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 51
+logminus:=rule(log(x + a) - log(- x - a) == log(-1))
+--R
+--I (6) log(x + a) - log(- x - a) + %P == log(- 1) + %P
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 52 14:170 Schaums and Axiom differ by a constant
+dd:=logminus cc
+--R
+--R log(- 1)
+--R (7) --------
+--R 3
+--R 4a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.171~~~~~$\displaystyle\int{\frac{x~dx}{(a^2-x^2)^2}}$}
@@ -164,7 +604,7 @@ $$
<<*>>=
)clear all
---S 9 of 19
+--S 53
aa:=integrate(x/((a^2-x^2)^2),x)
--R
--R
@@ -174,6 +614,24 @@ aa:=integrate(x/((a^2-x^2)^2),x)
--R 2x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 54
+bb:=1/(2*(a^2-x^2))
+--R
+--R 1
+--R (2) - ---------
+--R 2 2
+--R 2x - 2a
+--R Type: Fraction Polynomial
Integer
+--E
+
+--S 55 14:171 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.172~~~~~$\displaystyle\int{\frac{x^2dx}{(a^2-x^2)^2}}$}
@@ -183,7 +641,7 @@ $$
<<*>>=
)clear all
---S 10 of 19
+--S 56
aa:=integrate(x^2/((a^2-x^2)^2),x)
--R
--R
@@ -194,6 +652,63 @@ aa:=integrate(x^2/((a^2-x^2)^2),x)
--R 4a x - 4a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 57
+bb:=x/(2*(a^2-x^2))-1/(4*a)*log((a+x)/(a-x))
+--R
+--R 2 2 - x - a
+--R (- x + a )log(-------) - 2a x
+--R x - a
+--R (2) ------------------------------
+--R 2 3
+--R 4a x - 4a
+--R Type: Expression
Integer
+--E
+
+--S 58
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 59
+bb1:=divlog bb
+--R
+--R 2 2 2 2
+--R (x - a )log(x - a) + (- x + a )log(- x - a) - 2a x
+--R (4) ----------------------------------------------------
+--R 2 3
+--R 4a x - 4a
+--R Type: Expression
Integer
+--E
+
+--S 60
+cc:=aa-bb1
+--R
+--R - log(x + a) + log(- x - a)
+--R (5) ---------------------------
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 61
+logminus2:=rule(-log(x + a) + log(- x - a) == log(-1))
+--R
+--I (6) - log(x + a) + log(- x - a) + %S == log(- 1) + %S
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 62 14:172 Schaums and Axiom differ by a constant
+dd:=logminus2 cc
+--R
+--R log(- 1)
+--R (7) --------
+--R 4a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.173~~~~~$\displaystyle\int{\frac{x^3dx}{(a^2-x^2)^2}}$}
@@ -203,7 +718,7 @@ $$
<<*>>=
)clear all
---S 11 of 19
+--S 63
aa:=integrate(x^3/((a^2-x^2)^2),x)
--R
--R
@@ -214,6 +729,45 @@ aa:=integrate(x^3/((a^2-x^2)^2),x)
--R 2x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 64
+bb:=a^2/(2*(a^2-x^2))+1/2*log(a^2-x^2)
+--R
+--R 2 2 2 2 2
+--R (x - a )log(- x + a ) - a
+--R (2) ----------------------------
+--R 2 2
+--R 2x - 2a
+--R Type: Expression
Integer
+--E
+
+--S 65
+cc:=aa-bb
+--R
+--R 2 2 2 2
+--R log(x - a ) - log(- x + a )
+--R (3) -----------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 66
+logminus3:=rule(log(x^2-a^2)-log(-x^2+a^2) == log(-1))
+--R
+--R 2 2 2 2
+--I (4) log(x - a ) - log(- x + a ) + %T == log(- 1) + %T
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 67 14:173 Schaums and Axiom differ by a constant
+dd:=logminus3 cc
+--R
+--R log(- 1)
+--R (5) --------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.174~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^2}}$}
@@ -223,7 +777,7 @@ $$
<<*>>=
)clear all
---S 12 of 19
+--S 68
aa:=integrate(1/(x*(a^2-x^2)^2),x)
--R
--R
@@ -234,6 +788,70 @@ aa:=integrate(1/(x*(a^2-x^2)^2),x)
--R 2a x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 69
+bb:=1/(2*a^2*(a^2-x^2))+1/(2*a^4)*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R 2 2 x 2
+--R (x - a )log(- -------) - a
+--R 2 2
+--R x - a
+--R (2) ----------------------------
+--R 4 2 6
+--R 2a x - 2a
+--R Type: Expression
Integer
+--E
+
+--S 70
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 71
+bb1:=divlog bb
+--R
+--R 2 2 2 2 2 2 2 2
+--R (- x + a )log(x - a ) + (x - a )log(- x ) - a
+--R (4) -------------------------------------------------
+--R 4 2 6
+--R 2a x - 2a
+--R Type: Expression
Integer
+--E
+
+--S 72
+cc:=aa-bb1
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 73
+logpowminus:=rule(log(-a^n) == n*log(a)+log(-1))
+--R
+--R n
+--R (6) log(- a ) == n log(a) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 74 14:174 Schaums and Axiom differ by a constant
+dd:=logpowminus cc
+--R
+--R log(- 1)
+--R (7) - --------
+--R 4
+--R 2a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.175~~~~~$\displaystyle\int{\frac{dx}{x^2(a^2-x^2)^2}}$}
@@ -244,17 +862,88 @@ $$
<<*>>=
)clear all
---S 13 of 19
-aa:=integrate(1/((a^2-x^2)^2),x)
---R
+--S 75
+aa:=integrate(1/(x^2*(a^2-x^2)^2),x)
--R
---R 2 2 2 2
---R (x - a )log(x + a) + (- x + a )log(x - a) - 2a x
---R (1) --------------------------------------------------
---R 3 2 5
---R 4a x - 4a
+--R 3 2 3 2 2 3
+--R (3x - 3a x)log(x + a) + (- 3x + 3a x)log(x - a) - 6a x + 4a
+--R (1) ---------------------------------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 76
+bb:=-1/(a^4*x)+x/(2*a^4*(a^2-x^2))+3/(4*a^5)*log((a+x)/(a-x))
+--R
+--R 3 2 - x - a 2 3
+--R (3x - 3a x)log(-------) - 6a x + 4a
+--R x - a
+--R (2) --------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
+--R Type: Expression
Integer
+--E
+
+--S 77
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 78
+bb1:=divlog bb
+--R
+--R 3 2 3 2 2 3
+--R (- 3x + 3a x)log(x - a) + (3x - 3a x)log(- x - a) - 6a x + 4a
+--R (4) -----------------------------------------------------------------
+--R 5 3 7
+--R 4a x - 4a x
+--R Type: Expression
Integer
+--E
+
+--S 79
+cc:=aa-bb
+--R
+--R - x - a
+--R 3log(x + a) - 3log(x - a) - 3log(-------)
+--R x - a
+--R (5) -----------------------------------------
+--R 5
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 80
+dd:=divlog cc
+--R
+--R 3log(x + a) - 3log(- x - a)
+--R (6) ---------------------------
+--R 5
+--R 4a
+--R Type: Expression
Integer
+--E
+
+--S 81
+logminusb:=rule(b*log(x + a) - b*log(- x - a) == b*log(-1))
+--R
+--I (7) b log(x + a) - b log(- x - a) + %U == b log(- 1) + %U
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 82 14:175 Schaums and Axiom differ by a constant
+ee:=logminusb dd
+--R
+--R 3log(- 1)
+--R (8) ---------
+--R 5
+--R 4a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.176~~~~~$\displaystyle\int{\frac{dx}{x^3(a^2-x^2)^2}}$}
@@ -265,7 +954,7 @@ $$
<<*>>=
)clear all
---S 14 of 19
+--S 83
aa:=integrate(1/(x^3*(a^2-x^2)^2),x)
--R
--R
@@ -276,6 +965,70 @@ aa:=integrate(1/(x^3*(a^2-x^2)^2),x)
--R 2a x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 84
+bb:=-1/(2*a^4*x^2)+1/(2*a^4*(a^2-x^2))+1/a^6*log(x^2/(a^2-x^2))
+--R
+--R 2
+--R 4 2 2 x 2 2 4
+--R (2x - 2a x )log(- -------) - 2a x + a
+--R 2 2
+--R x - a
+--R (2) ----------------------------------------
+--R 6 4 8 2
+--R 2a x - 2a x
+--R Type: Expression
Integer
+--E
+
+--S 85
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (3) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 86
+bb1:=divlog bb
+--R
+--R 4 2 2 2 2 4 2 2 2 2 2 4
+--R (- 2x + 2a x )log(x - a ) + (2x - 2a x )log(- x ) - 2a x + a
+--R (4) -----------------------------------------------------------------
+--R 6 4 8 2
+--R 2a x - 2a x
+--R Type: Expression
Integer
+--E
+
+--S 87
+cc:=aa-bb1
+--R
+--R 2
+--R 2log(x) - log(- x )
+--R (5) -------------------
+--R 6
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 88
+logpowminus:=rule(log(-a^n) == n*log(a)+log(-1))
+--R
+--R n
+--R (6) log(- a ) == n log(a) + log(- 1)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 89 14:176 Schaums and Axiom differ by a constant
+dd:=logpowminus cc
+--R
+--R log(- 1)
+--R (7) - --------
+--R 6
+--R a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.177~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^n}}$}
@@ -286,7 +1039,7 @@ $$
<<*>>=
)clear all
---S 15 of 19
+--S 90 14:177 Axiom cannot do this integration
aa:=integrate(1/((a^2-x^2)^n),x)
--R
--R
@@ -306,7 +1059,7 @@ $$
<<*>>=
)clear all
---S 16 of 19
+--S 91
aa:=integrate(x/((a^2-x^2)^n),x)
--R
--R
@@ -318,6 +1071,49 @@ aa:=integrate(x/((a^2-x^2)^n),x)
--R (2n - 2)%e
--R Type: Union(Expression
Integer,...)
--E
+
+--S 92
+bb:=1/(2*(n-1)*(a^2-x^2)^(n-1))
+--R
+--R 1
+--R (2) ------------------------
+--R 2 2 n - 1
+--R (2n - 2)(- x + a )
+--R Type: Expression
Integer
+--E
+
+--S 93
+cc:=aa-bb
+--R
+--R 2 2
+--R n log(- x + a ) 2 2 2 2 n - 1
+--R - %e + (- x + a )(- x + a )
+--R (3) --------------------------------------------------
+--R 2 2
+--R 2 2 n - 1 n log(- x + a )
+--R (2n - 2)(- x + a ) %e
+--R Type: Expression
Integer
+--E
+
+--S 94
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R n log(x) n
+--R (4) %e == x
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 95 14:178 Axiom cannot simplify this expression
+dd:=explog cc
+--R
+--R 2 2 n 2 2 2 2 n - 1
+--R - (- x + a ) + (- x + a )(- x + a )
+--R (5) --------------------------------------------
+--R 2 2 n - 1 2 2 n
+--R (2n - 2)(- x + a ) (- x + a )
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$}
@@ -328,7 +1124,7 @@ $$
<<*>>=
)clear all
---S 17 of 19
+--S 96 14:179 Axiom cannot integrate this expression
aa:=integrate(1/(x*(a^2-x^2)^n),x)
--R
--R
@@ -349,7 +1145,7 @@ $$
<<*>>=
)clear all
---S 18 of 19
+--S 97 14:180 Axiom cannot integrate this expression
aa:=integrate(x^m/((a^2-x^2)^n),x)
--R
--R
@@ -370,7 +1166,7 @@ $$
<<*>>=
)clear all
---S 19 of 19
+--S 98 14:181 Axiom cannot integrate this expression
aa:=integrate(1/(x^m*(a^2-x^2)^n),x)
--R
--R
diff --git a/src/input/schaum9.input.pamphlet b/src/input/schaum9.input.pamphlet
index 6610811..fceb094 100644
--- a/src/input/schaum9.input.pamphlet
+++ b/src/input/schaum9.input.pamphlet
@@ -16,7 +16,7 @@ $$\int{\frac{1}{\sqrt{x^2+a^2}}}=\sinh^{-1}\frac{x}{a}$$
)set message auto off
)clear all
---S 1 of 28
+--S 1
aa:=integrate(1/(sqrt(x^2+a^2)),x)
--R
--R
@@ -25,14 +25,33 @@ aa:=integrate(1/(sqrt(x^2+a^2)),x)
--R (1) - log(\|x + a - x)
--R Type: Union(Expression
Integer,...)
--E
+
+--S 2
+bb:=log(x+sqrt(x^2+a^2))
+--R
+--R +-------+
+--R | 2 2
+--R (2) log(\|x + a + x)
+--R Type: Expression
Integer
+--E
+
+--S 3 14:182 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x + a + x) - log(\|x + a - x)
+--R Type: Expression
Integer
+--E
@
+This is equal to $-\log(a^2)$ but Axiom cannot prove it.
\section{\cite{1}:14.183~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2+a^2}}}$}
$$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$
<<*>>=
)clear all
---S 2 of 28
+--S 4
aa:=integrate(x/(sqrt(x^2+a^2)),x)
--R
--R
@@ -45,6 +64,22 @@ aa:=integrate(x/(sqrt(x^2+a^2)),x)
--R \|x + a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 5
+bb:=sqrt(x^2+a^2)
+--R
+--R +-------+
+--R | 2 2
+--R (2) \|x + a
+--R Type: Expression
Integer
+--E
+
+--S 6 14:183 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.184~~~~~$\displaystyle
@@ -55,7 +90,7 @@ $$
<<*>>=
)clear all
---S 3 of 28
+--S 7
aa:=integrate(x^2/sqrt(x^2+a^2),x)
--R
--R
@@ -73,6 +108,46 @@ aa:=integrate(x^2/sqrt(x^2+a^2),x)
--R 4x\|x + a - 4x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 8
+bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 | 2 2
+--R - a log(\|x + a + x) + x\|x + a
+--R (2) -------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 9
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R a log(\|x + a + x) + a log(\|x + a - x)
+--R (3) ---------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 10
+logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b))
+--R
+--I (4) c log(b) + c log(a) + %K == c log(a b) + %K
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 11 14:184 Schaums and Axiom differ by a constant
+dd:=logmul1 cc
+--R
+--R 2 2
+--R a log(a )
+--R (5) ---------
+--R 2
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.185~~~~~$\displaystyle
@@ -83,7 +158,7 @@ $$
<<*>>=
)clear all
---S 4 of 28
+--S 12
aa:=integrate(x^3/sqrt(x^2+a^2),x)
--R
--R
@@ -96,6 +171,24 @@ aa:=integrate(x^3/sqrt(x^2+a^2),x)
--R (12x + 3a )\|x + a - 12x - 9a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 13
+bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2)
+--R
+--R +-------+
+--R 2 2 | 2 2
+--R (x - 2a )\|x + a
+--R (2) --------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 14 14:185 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.186~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2+a^2}}}$}
@@ -105,7 +198,7 @@ $$
<<*>>=
)clear all
---S 5 of 28
+--S 15
aa:=integrate(1/(x*sqrt(x^2+a^2)),x)
--R
--R
@@ -116,6 +209,57 @@ aa:=integrate(1/(x*sqrt(x^2+a^2)),x)
--R a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 16
+bb:=-1/a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2
+--R \|x + a + a
+--R log(--------------)
+--R x
+--R (2) - -------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 17
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+
+--R +-------+ +-------+ | 2 2
+--R | 2 2 | 2 2 \|x + a + a
+--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------)
+--R x
+--R -------------------------------------------------------------------------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 18
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 19 14:186 Axiom cannot simplify this expression
+dd:=divlog cc
+--R
+--R (5)
+--R +-------+ +-------+ +-------+
+--R | 2 2 | 2 2 | 2 2
+--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x -
a)
+--R +
+--R - log(x)
+--R /
+--R a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.187~~~~~$\displaystyle
@@ -126,7 +270,7 @@ $$
<<*>>=
)clear all
---S 6 of 28
+--S 20
aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x)
--R
--R
@@ -137,6 +281,29 @@ aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x)
--R x\|x + a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 21
+bb:=-sqrt(x^2+a^2)/(a^2*x)
+--R
+--R +-------+
+--R | 2 2
+--R \|x + a
+--R (2) - ----------
+--R 2
+--R a x
+--R Type: Expression
Integer
+--E
+
+--S 22 14:187 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 1
+--R (3) - --
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.188~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2+a^2}}}$}
@@ -147,7 +314,7 @@ $$
<<*>>=
)clear all
---S 7 of 28
+--S 23
aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x)
--R
--R
@@ -169,6 +336,59 @@ aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x)
--R 4a x \|x + a - 4a x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2 +-------+
+--R 2 \|x + a + a | 2 2
+--R x log(--------------) - a\|x + a
+--R x
+--R (2) -----------------------------------
+--R 3 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 25
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+
+--R +-------+ +-------+ | 2 2
+--R | 2 2 | 2 2 \|x + a + a
+--R log(\|x + a - x + a) - log(\|x + a - x - a) - log(--------------)
+--R x
+--R -----------------------------------------------------------------------
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 26
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 27 14:188 Axiom cannot simplify this expression
+dd:=divlog cc
+--R
+--R (5)
+--R +-------+ +-------+ +-------+
+--R | 2 2 | 2 2 | 2 2
+--R - log(\|x + a + a) + log(\|x + a - x + a) - log(\|x + a - x
- a)
+--R +
+--R log(x)
+--R /
+--R 3
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.189~~~~~$\displaystyle\int{\sqrt{x^2+a^2}}~dx$}
@@ -178,7 +398,7 @@ $$
<<*>>=
)clear all
---S 8 of 28
+--S 28
aa:=integrate(sqrt(x^2+a^2),x)
--R
--R
@@ -196,6 +416,28 @@ aa:=integrate(sqrt(x^2+a^2),x)
--R 4x\|x + a - 4x - 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 29
+bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 | 2 2
+--R a log(\|x + a + x) + x\|x + a
+--R (2) -----------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 30 14:189 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2
+--R - a log(\|x + a + x) - a log(\|x + a - x)
+--R (3) -----------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.190~~~~~$\displaystyle\int{x\sqrt{x^2+a^2}}~dx$}
@@ -205,7 +447,7 @@ $$
<<*>>=
)clear all
---S 9 of 28
+--S 31
aa:=integrate(x*sqrt(x^2+a^2),x)
--R
--R
@@ -218,6 +460,24 @@ aa:=integrate(x*sqrt(x^2+a^2),x)
--R (12x + 3a )\|x + a - 12x - 9a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb:=(x^2+a^2)^(3/2)/3
+--R
+--R +-------+
+--R 2 2 | 2 2
+--R (x + a )\|x + a
+--R (2) -------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 33 14:190 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.191~~~~~$\displaystyle
@@ -229,7 +489,7 @@ $$
<<*>>=
)clear all
---S 10 of 28
+--S 34
aa:=integrate(x^2*sqrt(x^2+a^2),x)
--R
--R
@@ -247,6 +507,28 @@ aa:=integrate(x^2*sqrt(x^2+a^2),x)
--R (64x + 32a x)\|x + a - 64x - 64a x - 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 35
+bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 3 2 | 2 2
+--R - a log(\|x + a + x) + (2x + a x)\|x + a
+--R (2) -----------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 36 14:191 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R a log(\|x + a + x) + a log(\|x + a - x)
+--R (3) ---------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.192~~~~~$\displaystyle
@@ -257,7 +539,7 @@ $$
<<*>>=
)clear all
---S 11 of 28
+--S 37
aa:=integrate(x^3*sqrt(x^2+a^2),x)
--R
--R
@@ -274,6 +556,24 @@ aa:=integrate(x^3*sqrt(x^2+a^2),x)
--R (240x + 180a x + 15a )\|x + a - 240x - 300a x - 75a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 38
+bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3
+--R
+--R +-------+
+--R 4 2 2 4 | 2 2
+--R (3x + a x - 2a )\|x + a
+--R (2) ----------------------------
+--R 15
+--R Type: Expression
Integer
+--E
+
+--S 39 14:192 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.193~~~~~$\displaystyle
@@ -284,7 +584,7 @@ $$
<<*>>=
)clear all
---S 12 of 28
+--S 40
aa:=integrate(sqrt(x^2+a^2)/x,x)
--R
--R
@@ -302,6 +602,33 @@ aa:=integrate(sqrt(x^2+a^2)/x,x)
--R \|x + a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 41
+bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2 +-------+
+--R \|x + a + a | 2 2
+--R (2) - a log(--------------) + \|x + a
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 42 14:193 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - a log(\|x + a - x + a) + a log(\|x + a - x - a)
+--R +
+--R +-------+
+--R | 2 2
+--R \|x + a + a
+--R a log(--------------)
+--R x
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.194~~~~~$\displaystyle
@@ -312,7 +639,7 @@ $$
<<*>>=
)clear all
---S 13 of 28
+--S 43
aa:=integrate(sqrt(x^2+a^2)/x^2,x)
--R
--R
@@ -325,6 +652,26 @@ aa:=integrate(sqrt(x^2+a^2)/x^2,x)
--R x\|x + a - x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 44
+bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R x log(\|x + a + x) - \|x + a
+--R (2) ----------------------------------
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 45 14:194 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.195~~~~~$\displaystyle
@@ -336,7 +683,7 @@ $$
<<*>>=
)clear all
---S 14 of 28
+--S 46
aa:=integrate(sqrt(x^2+a^2)/x^3,x)
--R
--R
@@ -358,6 +705,57 @@ aa:=integrate(sqrt(x^2+a^2)/x^3,x)
--R 4a x \|x + a - 4a x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 47
+bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2 +-------+
+--R 2 \|x + a + a | 2 2
+--R - x log(--------------) - a\|x + a
+--R x
+--R (2) -------------------------------------
+--R 2
+--R 2a x
+--R Type: Expression
Integer
+--E
+
+--S 48
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+
+--R +-------+ +-------+ | 2 2
+--R | 2 2 | 2 2 \|x + a + a
+--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------)
+--R x
+--R -------------------------------------------------------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 49
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 50 14:195 Axiom cannot simplify this expression
+dd:=divlog cc
+--R
+--R (5)
+--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 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.196~~~~~$\displaystyle\int{\frac{dx}{(x^2+a^2)^{3/2}}}$}
@@ -367,7 +765,7 @@ $$
<<*>>=
)clear all
---S 15 of 28
+--S 51
aa:=integrate(1/(x^2+a^2)^(3/2),x)
--R
--R
@@ -378,6 +776,27 @@ aa:=integrate(1/(x^2+a^2)^(3/2),x)
--R x\|x + a - x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 52
+bb:=x/(a^2*sqrt(x^2+a^2))
+--R
+--R x
+--R (2) ------------
+--R +-------+
+--R 2 | 2 2
+--R a \|x + a
+--R Type: Expression
Integer
+--E
+
+--S 53 14:196 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 1
+--R (3) --
+--R 2
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.197~~~~~$\displaystyle
@@ -388,7 +807,7 @@ $$
<<*>>=
)clear all
---S 16 of 28
+--S 54
aa:=integrate(x/(x^2+a^2)^(3/2),x)
--R
--R
@@ -401,6 +820,25 @@ aa:=integrate(x/(x^2+a^2)^(3/2),x)
--R x\|x + a - x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=-1/sqrt(x^2+a^2)
+--R
+--R 1
+--R (2) - ----------
+--R +-------+
+--R | 2 2
+--R \|x + a
+--R Type: Expression
Integer
+--E
+
+--S 56 14:197 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
+
@
\section{\cite{1}:14.198~~~~~$\displaystyle
@@ -411,7 +849,7 @@ $$
<<*>>=
)clear all
---S 17 of 28
+--S 57
aa:=integrate(x^2/(x^2+a^2)^(3/2),x)
--R
--R
@@ -424,6 +862,28 @@ aa:=integrate(x^2/(x^2+a^2)^(3/2),x)
--R x\|x + a - x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 58
+bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R \|x + a log(\|x + a + x) - x
+--R (2) ---------------------------------
+--R +-------+
+--R | 2 2
+--R \|x + a
+--R Type: Expression
Integer
+--E
+
+--S 59 14:198 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R (3) - log(\|x + a + x) - log(\|x + a - x) - 1
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.199~~~~~$\displaystyle
@@ -434,7 +894,7 @@ $$
<<*>>=
)clear all
---S 18 of 28
+--S 60
aa:=integrate(x^3/(x^2+a^2)^(3/2),x)
--R
--R
@@ -447,6 +907,25 @@ aa:=integrate(x^3/(x^2+a^2)^(3/2),x)
--R (2x + a )\|x + a - 2x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 61
+bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2)
+--R
+--R 2 2
+--R x + 2a
+--R (2) ----------
+--R +-------+
+--R | 2 2
+--R \|x + a
+--R Type: Expression
Integer
+--E
+
+--S 62 14:199 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.200~~~~~$\displaystyle
@@ -458,7 +937,7 @@ $$
<<*>>=
)clear all
---S 19 of 28
+--S 63
aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x)
--R
--R
@@ -476,6 +955,60 @@ aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x)
--R a x\|x + a - a x - a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 64
+bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R | 2 2 \|x + a + a
+--R - \|x + a log(--------------) + a
+--R x
+--R (2) -----------------------------------
+--R +-------+
+--R 3 | 2 2
+--R a \|x + a
+--R Type: Expression
Integer
+--E
+
+--S 65
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+
+--R +-------+ +-------+ | 2 2
+--R | 2 2 | 2 2 \|x + a + a
+--R - log(\|x + a - x + a) + log(\|x + a - x - a) + log(--------------)
+--R x
+--R -------------------------------------------------------------------------
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 66
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 67 14:200 Axiom cannot simplify this expression
+dd:=divlog cc
+--R
+--R (5)
+--R +-------+ +-------+ +-------+
+--R | 2 2 | 2 2 | 2 2
+--R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x -
a)
+--R +
+--R - log(x)
+--R /
+--R 3
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.201~~~~~$\displaystyle
@@ -486,7 +1019,7 @@ $$
<<*>>=
)clear all
---S 20 of 28
+--S 68
aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x)
--R
--R
@@ -497,6 +1030,28 @@ aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x)
--R (2x + a x)\|x + a - 2x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 69
+bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2))
+--R
+--R 2 2
+--R - 2x - a
+--R (2) -------------
+--R +-------+
+--R 4 | 2 2
+--R a x\|x + a
+--R Type: Expression
Integer
+--E
+
+--S 70 14:201 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R 2
+--R (3) - --
+--R 4
+--R a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.202~~~~~$\displaystyle
@@ -509,7 +1064,7 @@ $$
<<*>>=
)clear all
---S 21 of 28
+--S 71
aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x)
--R
--R
@@ -535,6 +1090,62 @@ aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x)
--R (8a x + 6a x )\|x + a - 8a x - 10a x - 2a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 72
+bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R +-------+ | 2 2
+--R 2 | 2 2 \|x + a + a 2 3
+--R 3x \|x + a log(--------------) - 3a x - a
+--R x
+--R (2) ---------------------------------------------
+--R +-------+
+--R 5 2 | 2 2
+--R 2a x \|x + a
+--R Type: Expression
Integer
+--E
+
+--S 73
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+
+--R +-------+ +-------+ | 2 2
+--R | 2 2 | 2 2 \|x + a + a
+--R 3log(\|x + a - x + a) - 3log(\|x + a - x - a) -
3log(--------------)
+--R x
+--R
--------------------------------------------------------------------------
+--R 5
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 74
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 75 14:202 Axiom cannot simplify this expression
+dd:=divlog cc
+--R
+--R (5)
+--R +-------+ +-------+
+--R | 2 2 | 2 2
+--R - 3log(\|x + a + a) + 3log(\|x + a - x + a)
+--R +
+--R +-------+
+--R | 2 2
+--R - 3log(\|x + a - x - a) + 3log(x)
+--R /
+--R 5
+--R 2a
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.203~~~~~$\displaystyle\int{(x^2+a^2)^{3/2}}~dx$}
@@ -545,9 +1156,8 @@ $$
<<*>>=
)clear all
---S 22 of 28
+--S 76
aa:=integrate((x^2+a^2)^(3/2),x)
---R
--R
--R (1)
--R +-------+ +-------+
@@ -566,6 +1176,28 @@ aa:=integrate((x^2+a^2)^(3/2),x)
--R (64x + 32a x)\|x + a - 64x - 64a x - 8a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 77
+bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 3 2 | 2 2
+--R 3a log(\|x + a + x) + (2x + 5a x)\|x + a
+--R (2) -----------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
+
+--S 78 14:203 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 4 | 2 2 4 | 2 2
+--R - 3a log(\|x + a + x) - 3a log(\|x + a - x)
+--R (3) -------------------------------------------------
+--R 8
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.204~~~~~$\displaystyle\int{x(x^2+a^2)^{3/2}}~dx$}
@@ -573,7 +1205,7 @@ $$\int{x(x^2+a^2)^{3/2}}=\frac{(x^2+a^2)^{5/2}}{5}$$
<<*>>=
)clear all
---S 23 of 28
+--S 79
aa:=integrate(x*(x^2+a^2)^(3/2),x)
--R
--R
@@ -590,6 +1222,24 @@ aa:=integrate(x*(x^2+a^2)^(3/2),x)
--R (80x + 60a x + 5a )\|x + a - 80x - 100a x - 25a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 80
+bb:=(x^2+a^2)^(5/2)/5
+--R
+--R +-------+
+--R 4 2 2 4 | 2 2
+--R (x + 2a x + a )\|x + a
+--R (2) ---------------------------
+--R 5
+--R Type: Expression
Integer
+--E
+
+--S 81 14:204 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.205~~~~~$\displaystyle\int{x^2(x^2+a^2)^{3/2}}~dx$}
@@ -601,7 +1251,7 @@ $$
<<*>>=
)clear all
---S 24 of 28
+--S 82
aa:=integrate(x^2*(x^2+a^2)^(3/2),x)
--R
--R
@@ -629,6 +1279,28 @@ aa:=integrate(x^2*(x^2+a^2)^(3/2),x)
--R (1536x + 1536a x + 288a x)\|x + a - 1536x - 2304a x - 864a x -
48a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 83
+bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^6/16*log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R 6 | 2 2 5 2 3 4 | 2 2
+--R - 3a log(\|x + a + x) + (8x + 14a x + 3a x)\|x + a
+--R (2) ----------------------------------------------------------
+--R 48
+--R Type: Expression
Integer
+--E
+
+--S 84 14:205 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 6 | 2 2 6 | 2 2
+--R a log(\|x + a + x) + a log(\|x + a - x)
+--R (3) ---------------------------------------------
+--R 16
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.206~~~~~$\displaystyle\int{x^3(x^2+a^2)^{3/2}}~dx$}
@@ -638,7 +1310,7 @@ $$
<<*>>=
)clear all
---S 25 of 28
+--S 85
aa:=integrate(x^3*(x^2+a^2)^(3/2),x)
--R
--R
@@ -667,6 +1339,24 @@ aa:=integrate(x^3*(x^2+a^2)^(3/2),x)
--R - 1960a x - 245a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 86
+bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5
+--R
+--R +-------+
+--R 6 2 4 4 2 6 | 2 2
+--R (5x + 8a x + a x - 2a )\|x + a
+--R (2) ------------------------------------
+--R 35
+--R Type: Expression
Integer
+--E
+
+--S 87 14:206 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R (3) 0
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.207~~~~~$\displaystyle
@@ -678,7 +1368,7 @@ $$
<<*>>=
)clear all
---S 26 of 28
+--S 88
aa:=integrate((x^2+a^2)^(3/2)/x,x)
--R
--R
@@ -700,6 +1390,58 @@ aa:=integrate((x^2+a^2)^(3/2)/x,x)
--R (12x + 3a )\|x + a - 12x - 9a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 89
+bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2 +-------+
+--R 3 \|x + a + a 2 2 | 2 2
+--R - 3a log(--------------) + (x + 4a )\|x + a
+--R x
+--R (2) -----------------------------------------------
+--R 3
+--R Type: Expression
Integer
+--E
+
+--S 90
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+ +-------+
+--R 3 | 2 2 3 | 2 2
+--R - a log(\|x + a - x + a) + a log(\|x + a - x - a)
+--R +
+--R +-------+
+--R | 2 2
+--R 3 \|x + a + a
+--R a log(--------------)
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 91
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R a
+--R (4) log(-) == - log(b) + log(a)
+--R b
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 92 14:207 Axiom cannot simplify this expression
+dd:=divlog cc
+--R
+--R (5)
+--R +-------+ +-------+
+--R 3 | 2 2 3 | 2 2
+--R a log(\|x + a + a) - a log(\|x + a - x + a)
+--R +
+--R +-------+
+--R 3 | 2 2 3
+--R a log(\|x + a - x - a) - a log(x)
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.208~~~~~$\displaystyle
@@ -711,7 +1453,7 @@ $$
<<*>>=
)clear all
---S 27 of 28
+--S 93
aa:=integrate((x^2+a^2)^{3/2}/x^2,x)
--R
--R
@@ -729,6 +1471,28 @@ aa:=integrate((x^2+a^2)^{3/2}/x^2,x)
--R (8x + 2a x)\|x + a - 8x - 6a x
--R Type: Union(Expression
Integer,...)
--E
+
+--S 94
+bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2))
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 2 | 2 2
+--R 3a x log(\|x + a + x) + (x - 2a )\|x + a
+--R (2) -----------------------------------------------
+--R 2x
+--R Type: Expression
Integer
+--E
+
+--S 95 14:208 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R +-------+ +-------+
+--R 2 | 2 2 2 | 2 2 2
+--R - 3a log(\|x + a + x) - 3a log(\|x + a - x) - 2a
+--R (3) -------------------------------------------------------
+--R 2
+--R Type: Expression
Integer
+--E
@
\section{\cite{1}:14.209~~~~~$\displaystyle
@@ -740,7 +1504,7 @@ $$
<<*>>=
)clear all
---S 28 of 28
+--S 96
aa:=integrate((x^2+a^2)^(3/2)/x^3,x)
--R
--R
@@ -763,6 +1527,51 @@ aa:=integrate((x^2+a^2)^(3/2)/x^3,x)
--R Type: Union(Expression
Integer,...)
--E
+--S 97
+bb:=-(x^2+a^2)^(3/2)/(2*x^3)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x)
+--R
+--R +-------+
+--R | 2 2 +-------+
+--R 3 \|x + a + a 3 2 2 | 2 2
+--R - 3a x log(--------------) + (3x - x - a )\|x + a
+--R x
+--R (2) ------------------------------------------------------
+--R 3
+--R 2x
+--R Type: Expression
Integer
+--E
+
+--S 98 14:209 Axiom cannot simplify this expression
+cc:=aa-bb
+--R
+--R (3)
+--R +-------+ +-------+
+--R 5 3 3 | 2 2 6 3 4 | 2 2
+--R ((- 12a x - 3a x )\|x + a + 12a x + 9a x )log(\|x + a - x +
a)
+--R +
+--R +-------+ +-------+
+--R 5 3 3 | 2 2 6 3 4 | 2 2
+--R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(\|x + a - x - a)
+--R +
+--R +-------+
+--R +-------+ | 2 2
+--R 5 3 3 | 2 2 6 3 4 \|x + a + a
+--R ((12a x + 3a x )\|x + a - 12a x - 9a x )log(--------------)
+--R x
+--R +
+--R +-------+
+--R 6 5 2 4 2 3 4 2 4 | 2 2 7 6
2 5
+--R (4x - 4x + 7a x - 7a x + 3a x - 3a x)\|x + a - 4x + 4x -
9a x
+--R +
+--R 2 4 4 3 4 2 6 6
+--R 9a x - 6a x + 6a x - a x + a
+--R /
+--R +-------+
+--R 5 2 3 | 2 2 6 2 4
+--R (8x + 2a x )\|x + a - 8x - 6a x
+--R Type: Expression
Integer
+--E
+
)spool
)lisp (bye)
@
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- [Axiom-developer] 20080417.02.tpd.patch (CATS Schaums-Axiom equivalence testing (8-10)),
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