If somebody with Mathematica / Axiom / ... could try:
Integrate[(2*x-1)*sqrt[-(x^2-x-1)*(x^2-x+1)]/((x^2-x-1)*(x^2-x+1)),x],
I would appreciate it. Wolfram | Alpha doesn't get the antiderivative
and Maple 10 gives a huge expression involving various radicals
(sqrt(3) and sqrt(5)) and several inverse Jacobi functions. At least
for a naive Maple user (me), Maple isn't able to simplify the expression.
But the antiderivative isn't all that messy. (My) function elliptic_int
looks for antiderivatives that involve inverse_jacobi_sn. The
ev(diff(%,x),diff, ratsimp) crazyness is due to noun/verb confusion in the
simplification of inverse_jacobi_sn (the asin in %o221 is generated by
simplifying
a inverse_jacobi_sn expression).
(%i220) e : ((2*x-1)*sqrt(-(x^2-x-1)*(x^2-x+1)))/((x^2-x-1)*(x^2-x+1));
(%o220) ((2*x-1)*sqrt((-x^2+x+1)*(x^2-x+1)))/((x^2-x-1)*(x^2-x+1))
(%i221) elliptic_int(e,x);
(%o221) (%i*(2*x^4-4*x^3+2*x^2-2)*asin((sqrt(2)*%i*sqrt(x^2-x-1))/2))/(sqrt
(x^2-x-1)*sqrt(x^2-x+1)*sqrt(-x^4+2*x^3-x^2+1))
(%i222) ev(diff(%,x),diff, ratsimp);
(%o222) ((2*x-1)*sqrt(-x^4+2*x^3-x^2+1))/(x^4-2*x^3+x^2-1)
(%i223) scanmap('factor,%);
(%o223) ((2*x-1)*sqrt(-(x^2-x-1)*(x^2-x+1)))/((x^2-x-1)*(x^2-x+1))
Barton