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I have the following function $$E(\theta)=\int \frac{d\theta}{\sqrt{(1-\alpha\cos\theta)(1+\beta\cos\theta)}},$$ I cannot find a suitable change of variable such that $E(\theta)$ converted into elliptic integral.

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  • $\begingroup$ try $u = \tan(\theta / 2)$ $\endgroup$ – Youem Apr 8 '18 at 23:30
  • $\begingroup$ @Youem: Such comment is not very constructive. What is actually achieved through such substitution? $\endgroup$ – Jack D'Aurizio Apr 8 '18 at 23:40
  • $\begingroup$ Your claim is totaly correct. But my comment was just a hint to the OP. $\endgroup$ – Youem Apr 8 '18 at 23:42
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    $\begingroup$ after the $u = \tan\frac{\theta}{2}$ substitution, you need another one $\tan\psi = \sqrt{\frac{1+\alpha}{1-\alpha}}u$ $\endgroup$ – achille hui Apr 9 '18 at 1:11
  • $\begingroup$ This change of variable give me $\int\frac{u du}{\sqrt{[(1-a)+(1+a)u^2][(1+b)+(1-b)u^2]}}$ $\endgroup$ – user41512 Apr 11 '18 at 23:08

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