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, 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

> 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

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