Strange that Wolfram Alpha doesn't get it, because integrals.wolfram.com
does<http://integrals.wolfram.com/index.jsp?expr=%282*x-1%29*sqrt%5B-%28x%5E2-x-1%29*%28x%5E2-x%2B1%29%5D%2F%28%28x%5E2-x-1%29*%28x%5E2-x%2B1%29%29&random=false>;,
in terms of elliptic integrals:
*Integrate[ ((2*x - 1)*Sqrt[-((x^2 - x - 1)*(x^2 - x + 1))])/ ((x^2 - x -
1)*(x^2 - x + 1)), x] ==*
(3*(-1)^(1/6)*Sqrt[2]*((-I)*Sqrt[3] + Sqrt[5])* Sqrt[((-I)*(Sqrt[3] -
I*Sqrt[5])*(1 + Sqrt[5] - 2*x)*(-(-1)^(1/3) + x))/((I + Sqrt[15])*
((-1)^(2/3) + x)^2)]*((-1)^(2/3) + x)^2* Sqrt[(-1 + Sqrt[5] + 2*x)/((Sqrt[3]
- I*Sqrt[5])* ((-1)^(2/3) + x))]* (EllipticF[ArcSin[ Sqrt[-((3*I + Sqrt[3] -
I*Sqrt[5] + Sqrt[15] - 2*Sqrt[3]*x + (2*I)*Sqrt[5]*x)/ (-3*I + Sqrt[3] +
I*Sqrt[5] + Sqrt[15] - 2*(Sqrt[3] + I*Sqrt[5])*x))]], (I + Sqrt[15])/(I -
Sqrt[15])] - 2*EllipticPi[-((Sqrt[3] + I*Sqrt[5])/ (Sqrt[3] - I*Sqrt[5])),
ArcSin[Sqrt[-((3*I + Sqrt[3] - I*Sqrt[5] + Sqrt[15] - 2*Sqrt[3]*x +
(2*I)*Sqrt[5]*x)/ (-3*I + Sqrt[3] + I*Sqrt[5] + Sqrt[15] - 2*(Sqrt[3] +
I*Sqrt[5])*x))]], (I + Sqrt[15])/(I - Sqrt[15])]))/ ((1 +
(-1)^(1/3))*(I*Sqrt[3] + Sqrt[5])* Sqrt[1 - x^2 + 2*x^3 - x^4])
If you translate this into Maxima notation, be careful to set
domain:complex; otherwise you get a division by zero because (-1)^(1/3) =>
-1.
-s
On Tue, Jul 14, 2009 at 8:10 AM, Barton Willis <willisb at unk.edu> 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))
>
> (%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
>
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