Albert Rich schrieb: > > 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... >
Should be no problem once Rubi can be run without Mathematica (Warning: when I work on something like this, it is likely to accumulate changes everywhere). Actually, the more special a function, the fewer its integration rules; so what remains to be done on cylinder (Bessel) and parabolic cylinder functions, and on confluent and generalized hypergeometric and Meijer-G functions is pretty limited; Jacobian elliptic functions and friends may be the hardest part (cf. Labahn-Humphries 2005). Possibilities multiply, however, when one restricts to definite integrals over special intervals, such as the positive real semiaxis - but this is a separate story.
> 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 >
I haven't taken a close look yet but hope this is not what you called a "messy algorithm  not appropriate for an elegant, rule-based system;  it requires the host CAS provide a strong limit package and good algebraic simplification." Such requirements would severely limit the portability of Rubi.