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Topic: Goursat pseudo-elliptics and the Wolfram Integrator
Replies: 5   Last Post: Dec 18, 2012 12:15 PM

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Nasser Abbasi

Posts: 6,677
Registered: 2/7/05
Re: Goursat pseudo-elliptics and the Wolfram Integrator
Posted: Dec 15, 2012 3:08 AM
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On 12/14/2012 3:47 PM, wrote:
> I was curious how Mathematica would handle the pseudo-elliptic integrals
> recently referred to on this newsgroup. So I plugged two examples into
> the Wolfram Integrator (which presumably represents Mathematica 8):
> <>
> Example 1 (cubic radicand):
> Integrate[(k*x^2 - 1)/((a*k*x + b)*(b*x + a)
> *Sqrt[x*(1 - x)*(1 - k*x)]), x]
> ... eeeek! The Integrator replies in terms of incomplete elliptic F,
> incomplete elliptic Pi, and the imaginary unit. But the antiderivative
> just is:
> 2/(Sqrt[a*b]*Sqrt[(a + b)*(a*k + b)])
> *ArcTan[Sqrt[a*b]*Sqrt[x*(1 - x)*(1 - k*x)]
> /(Sqrt[(a + b)*(a*k + b)]*x)]

I tried this on version 9 of Mathematica. I get the same as with
the above web site you mention.

But when I plot the result for some values of a,b,k and compare with
what you have above, the plots do not look the same? Not even a scaled
shift, but they look different (near origin). Away from origin, they
seem to become closer to each others. (i.e. for very large x)
Here is the results to see:

I tried it on Maple16. Was not able to plot the result in
Maple, I get an empty plot. But Maple also gives result in
terms of incomplete elliptic functions also.

> Example 2 (quartic radicand):
> Integrate[(k*x^2 - 1)/((a*k*x + b)*(b*x + a)
> *Sqrt[(1 - x^2)*(1 - k^2*x^2)]), x]
> ... "Mathematica could not find a formula for your integral. Most likely
> this means that no formula exists." Waouw! Here the elementary
> antiderivative is:
> 2/(Sqrt[(a + b)*(a*k + b)]*Sqrt[(a - b)*(a*k - b)])
> *ArcTanh[Sqrt[(a + b)*(a*k + b)]*Sqrt[(1 - x^2)*(1 - k^2*x^2)]
> /(Sqrt[(a - b)*(a*k - b)]*(1 - x)*(1 - k*x))]

I tried this also, yes, Mathematica 9 does not do it.

Maple16 does solve it. It gives answers in terms of incomplete
elliptic as well. result is in the above link as well.

> The theory behind these integrals is given in: Edouard Goursat, Note sur
> quelques intégrales pseudo-elliptiques, Bulletin de la Société
> Mathématique de France 15 (1887), 106-120, on-line at:
> <>
> This was written 125 years ago - apparently too recent for the "Risch"
> integrator of Mathematica 8. I expect that FriCAS can do the second
> integral too. How do Maple and Sympy behave?
> Martin.

Do not use sympy, may be someone who has that can try them.


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