Barton Willis wrote:
>
> (%i126) hyper_int(1/sqrt(x*(x+1)*(x-1)),x);
> (%o126)
> (2*hypergeometric([1/4,1/2],[5/4],x^2)*x*sqrt(1-x^2))/sqrt(x^3-x)
>
> This might also be correct, but I don't know which recursions to use to
> show it.
> Evidence that %o126 might be correct:
>
> (%i127) taylor(diff(%,x) - 1/sqrt(x*(x+1)*(x-1)),x,0,128);
> (%o127)/T/ 0+...
>
It may be that this elliptic integral is particular because 0 is in the
middle of -1, 1, which is invariant by the homographies that preserve
infinity. And indeed:
(%i2) hyper_int(1/sqrt(x*(x+2)*(x-1)),x);
(%o2) false
while
(%i4) inverse_jacobi_int(1/sqrt(x*(x+1)*(x-1)),x);
(%o4)
%i sqrt(x - 1)
-2 %i inverse_jacobi_sn(--------------, 2) sqrt(x - 1) sqrt(x) sqrt(x + 1)
sqrt(2)
-------------------------------------------------------------------------
3
sqrt(x - x)
--
Michel Talon