Elliptic Integral-ish?

Question

I'm trying to solve this integral $\int (1-\cos(\theta))^{2}\sqrt{1-k^{2}cos(\theta)^{2}} \mathrm{d}\theta$

I think it's some kind of elliptic integral but i can't integrate.

Answer 1

Mathematica can integrate this. The code

Integrate[(1 - Cos[t])^2 Sqrt[1 - m Cos[t]^2], t] // FullSimplify // InputForm

returns

-(-12*(-1 + m)*Sqrt[m]*ArcTanh[(Sqrt[2]*Sqrt[m]*Sin[t])/
  Sqrt[2 - m - m*Cos[2*t]]]*Sqrt[2 - m - m*Cos[2*t]] +
4*(-1 + m)*(-1 + 5*m)*Sqrt[(-2 + m + m*Cos[2*t])/(-1 + m)]*
   EllipticE[t, m/(-1 + m)] +
4*(-1 + m)*Sqrt[(-2 + m + m*Cos[2*t])/
  (-1 + m)]*EllipticF[t, m/(-1 + m)] +
Sqrt[2]*m*(-2 + m + m*Cos[2*t])*(-6*Sin[t] + Sin[2*t])) /
  (12*m*Sqrt[2 - m - m*Cos[2*t]])

which can be written as $\, (A_1+A_2+A_3)/B\,$ where $\, u := m\cos(2t)+m-2\,$

$$ A_1 := -12 (1-m) \sqrt{m} \sqrt{u} \tan^{-1} \left(\frac{\sqrt{2} \sqrt{m} \sin (t)}{\sqrt{u}}\right), $$ $$ A_2 := -\frac{u \left(4 F\left(t\left|\frac{m}{m-1}\right.\right)+4 (5 m-1) E\left(t\left|\frac{m}{m-1}\right.\right)\right)}{\sqrt{\frac{u}{m-1}}}, $$ $$ A_3 := -\sqrt{2} m u (\sin (2 t)-6 \sin (t)),\quad B := 12m\sqrt{-u}. $$