Albert Rich schrieb: > > On Wednesday, November 13, 2013 1:56:55 AM UTC-10, clicl...@freenet.de wrote: > > > The hypergeometric series here (and in Example 3) terminate for a > > non-negative integer exponent n, while the series for your evaluation of > > Examples 6a,b terminate (after applying Euler's transformation) for a > > negative integer exponent n. The latter type of representation is > > somewhat more compact; a disadvantage is that the antiderivative 6a for > > positive a,m,x ends up on the branch cut of the hypergeometric function. > > I therefore propose to normalize to the former type: > > For the integral of (1+x)^n/x, Rubi currently returns > > -(1+x)^(1+n) * 2F1(1,1+n,2+n,1+x) / (1+n) > > In light of your comments above, would it be better to return > > (1+x)^n * 2F1(-n,-n,1-n,-1/x) / (n*(1+1/x)^n) ? >
I think so. For positive x and non-integer n you are no longer sitting right on the edge of a cliff then - the thought alone makes me dizzy. An equivalent (by Euler's transformation) but simpler antiderivative is:
(1+x)^(1+n) * 2F1(1,1,1-n,-1/x) / (n*x)
Note that the singularity at x=0 is already present in the integrand. And Pfaff's transformation of these two puts one on the brink of the chasm when x is negative and small:
(1+x)^n * 2F1(-n,1,1-n,1/(1+x)) / n
PS: I have seen your "piecewise constants" called "differential constants" by WRI's Daniel Lichtblau, but the former term is much much more popular according to Google.