I am stuck with a integration that is related to a pendulum problem.
$$ I = \frac{2\sqrt{2}}{\pi} \int_{x_1}^{x_2} \frac{\sqrt{-x^3+(1+E)x-\bar{\beta}}}{\sqrt{x} \sqrt{2-x^2}}dx\, $$
Note that, we have the following conditions: (1) $ 0<x_1<x_2\le \sqrt{2} $ ; (2) $ \bar{\beta}>0 $ , (3) $E$ can take any value
Someone told me that it would be easy to solve the integration via Contour Integration, but I cannot do it. Can someone help?
This was actually obtained by replacing, $1-\cos{q}=x^2$, in the following integral:
$$ I = \frac{\sqrt{2}}{\pi} \int_{q_1}^{q_2} \sqrt{E + \cos{q} -\frac{\beta}{\sqrt{2} \sqrt{1-\cos{q}}}}dq\, $$
First, I am not able to derive the branch points; Second, the limits are not standard and almost excludes all the poles. Therefore, I cannot proceed. Please help finding a way to do it.
If contour integration is not a way, can there be any luck with elliptic integrals? Or anything else?
Thanks in advance!!