Re: [Axiom-developer] Schaums help

[root]

>Doug Stewart wrote:
>> root wrote:
>>>>> My copy of Schaums (1968, printing 4) shows
>>>>>
>>>>> 14:334:
>>>>>
>>>>> int(1/(x*sqrt(x^n-a^n)),x) == 2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n))
>>>>>
>>>>> It seems this cannot be the answers.
>>>>> Can someone with a later version please check for a typo?
>>>>>
>>>>> Tim
>>>>>
>>>>>
>>>>> _______________________________________________
>>>>> Axiom-developer mailing list
>>>>> address@hidden
>>>>> http://lists.nongnu.org/mailman/listinfo/axiom-developer
>>>>>
>>>>>         
>>>> My schaums shows that answer.
>>>> also usind Maxima to do the derivative  I get the LHS.
>>>> (%i5) diff(2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n)),x);
>>>> (%o5) (a^n*x^(-n-1))/(sqrt(a^n)*sqrt(a^n/x^n)*sqrt(1-a^n/x^n))
>>>> (%i6) radcan(%);
>>>> (%o6) 1/(x*sqrt(x^n-a^n))
>>>>     
>>>
>>> If you compute
>>> aa:=integrate(1/(x*sqrt(x^n-a^n)),x)
>>> bb:=2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n))
>>> cc1:=aa.1-bb
>>> cc2:=aa.2-bb
>>>
>>> Can you find a simplification path (in Axiom) such that either cc1 or 
>>> cc2 simplify to a constant?
>>>
>>> Alternatively, can you use Maxima to find the constant?
>>>
>>> I'm failing to do either, although I'm still trying.
>>>
>>> Tim
>>>
>>>   
>>
>> Maxima seems to give a wrong answer for this integration.
>>
>>
>> (%i1) aa:integrate(1/(x*sqrt(x^n-a^n)),x);
>> Is  a  positive or negative?p;
>>
>> (%o1) (2*atan(sqrt(x^n-a^n)/a^(n/2)))/(a^(n/2)*n)
>>
>> and
>> (%i2) aa:integrate(1/(x*sqrt(x^n-a^n)),x);
>> Is  a  positive or negative?n;
>> (%o2) 
>> log((2*sqrt(x^n-a^n)-2*sqrt(-a^n))/(2*sqrt(x^n-a^n)+2*sqrt(-a^n)))/(sqrt(-a^n)*n)
>>  
>>
>>
>>
>> I'm not good enough at simplifying to help you.
>> Doug
>>
>>
>> _______________________________________________
>with help from Maxima
>> >Can you show me how to start with
>> >
>> >(2*atan(sqrt(x^n-a^n)/a^(n/2)))/(a^(n/2)*n)
>> >
>> >and differentiate it to get
>> >
>> >1/(x*sqrt(x^n-a^n)
>> >
>> >using Maxima.
>>   
>
>(%i4) (2*atan(sqrt(x^n-a^n)/a^(n/2)))/(a^(n/2)*n)$
>(%i5) ratsimp(diff(%,x));
>(%o5) 1/(x*sqrt(x^n-a^n))
>
>Barton
>
>so Maxima is not wrong.

None of these system are wrong and I believe that both expressions
are equal up to a constant.

What I'm looking for is a sequence of commands that will reduce
the difference of (Axiom's result)-(Schaums result) to that constant.

The subtle part is that there are many possible reduction paths
and the order of the transformations matter. It seems pretty
trivial to walk into a corner that will no longer reduce automatically.

Simplification is one of the hardest problems.

The problems in Schaums from 14.354 - 14.359 all seem to include 
the term:

   tan(%pi/4-(a*x)/2)

which seems to be a source of difficulty for the simplifications.
The original author had some insight that introduced this term
and I can't figure out what transformation I can use to reverse it.
The %pi/4 is clearly 45 degrees but I don't see many quarter-angle
reductions anywhere.

Tim