# How to transfer integral to elliptic one

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.

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