Barton Willis wrote:
> 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))
>
FWIW, here is a different antiderivative:
(%i115) -asin(x^2-x);
(%o115) -asin(x^2-x)
(%i116) ratsimp(e/diff(%,x));
(%o116) 1
I got this using my (currently broken) elliptic integral code. This
integral tickles a bug, but it gets far enough to show that the
substitution u=x-1/2 will reduce the integrand:
(%i118) ratsimp(subst([x=u+1/2],e));
(%o118) 8*u*sqrt(5-4*u^2)/((4*u^2-5)*sqrt(4*u^2+3))
If we eyeball this, this is the same as -8*u/sqrt((4*u^2+3)*(5-4*u^2))
(for appropriate range of u). Then
(%i119) ellintreduce(-8*u/y,u,y,5-4*u^2,4*u^2+3,z);
(%o119) 4*asin((1/2-2*u^2)/2)
There's another bug here, I think. The factor 4 shouldn't be there.
Substituting u=x-1/2 gives the above answer, once the factor 4 is removed.
Ray