I haven't found a Gauss hypergeometric function recursion that shows that hyper_int returns
a correct antiderivative. Here is a bug:
(%i1) (load(hyperint), load(orthopoly), domain : complex)$
(%i2) hyper_int(1/sqrt(x*(x+1)*(x-1)),x);
(%o2) (2*hypergeometric([1/4,1/2],[5/4],x^2)*x*sqrt(1-x^2))/sqrt(x^3-x)
(%i3) subst(hypergeometric = hgfred,%);
(%o3) (2*hgfred([1/4,1/2],[5/4],x^2)*x*sqrt(1-x^2))/sqrt(x^3-x)
(%i4) ev(%);
(%o4) (assoc_legendre_p(-1/4,-1/4,sqrt(1-x^2))*gamma(1/4)*x*sqrt(1-x^2))/(2^(3/4)*(x^2)^(1/8)*sqrt(x^3-x))
Wrong--should be 1/sqrt(x*(x+1)*(x-1))
(%i5) ratsimp(diff(%,x));
(%o5) 0
Could be an orthopoly bug, a hgfred bug, a hyper_int bug, a simplification bug, or.. Oh my.
--Barton