On 12/14/2012 3:47 PM, email@example.com 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): > > <http://integrals.wolfram.com/index.jsp> > > 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: > > <http://www.numdam.org/item?id=BSMF_1887__15__106_1> > > 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.