Date: Nov 21, 2013 1:11 AM
Author: Albert D. Rich
Subject: Re: The A. F. Timofeev symbolic integration test suite

On Saturday, November 16, 2013 2:47:27 AM UTC-10, clicl...@freenet.de wrote:

> Would the evaluations of Examples 3.n, 5a.n, 5b.n and 17 perhaps also
> profit from Euler's transformation?


Yes, I think Euler would approve of the optimal antiderivatives now shown for examples 3.n, 5a.n, 5b.n, 6a.n, 6b.n, 14 and 17 in the Timofeev Chapter 8 pdf file at

http://www.apmaths.uwo.ca/~arich/TimofeevChapter8TestResults.pdf

> I have rummaged my vaults and dug up a mildewed sheaf of papers with the
> gospel on 2F1 differentiation and integration. The integration part
> reads as follows (typed in without checking):
>
> INT(F21(a,b,c,x), x)
> = (c-1)/((a-1)*(b-1))*F21(a-1,b-1,c-1,x)
> [a/=1, b/=1, c/=1]
>
> INT(x^(b-2)*F21(a,b,c,x), x)
> = 1/(b-1)*x^(b-1)*F21(a,b-1,c,x)
> [b/=1]
>
> INT(x^(c-1)*F21(a,b,c,x), x)
> = 1/c*x^c*F21(a,b,c+1,x)
>
> INT((1-x)^(b-2)*F21(a,b,c,x), x)
> = (c-1)/((a-c+1)*(b-1))*(1-x)^(b-1)*F21(a,b-1,c-1,x)
> [b/=1, c-a/=1, c/=1]
>
> INT(x^(c-1)*(1-x)^(b-c-1)*F21(a,b,c,x), x)
> = 1/c*x^c*(1-x)^(b-c)*F21(a+1,b,c+1,x)
>
> This set should be augmented by applying Euler's transformation on both
> sides of each formula. Inasmuch as the 2F1 integration rules are unknown
> to Rubi, I suggest to implement the complete set. [...]


Rubi is an open-source project that needs contributions by others than me in order to reach its full potential. Also I am not an expert in hypergeometric functions and have no desire to become one. However, I would be delighted to incorporate a hypergeometric integration package written by someone knowledgeable in the field, like yourself...

> Looking forward to Rubi4.3forte,

Although not formally announced, Rubi 4.3 is now available for downloading at

http://www.apmaths.uwo.ca/~arich/

In addition to Euler's transformation it includes numerous improvements including the use of rectification to produce continuous antiderivatives after integrating trig expressions using the substitution u=tan(x) or u=tan(x/2). The algorithm is described in D.J.Jeffrey's 1997 paper "Rectifying Transformations for the Integration of Rational Trigonometric Functions" available at

http://www.apmaths.uwo.ca/~djeffrey/Offprints/trig-rec.pdf

Albert