I am wondering if there is any hope this indefinite integral
$$\int\frac{\cos^2(x)}{\sin^2(x)}\frac{x}{\sqrt{a^2-b^2\sin^4(x)}}dx$$
could be calculated analytically? It's actually a simplified version of a more complicated integral I'm dealing with. I've tried a number of substitutions, and integration by part only seems to make things worse. I also played with the Leibniz rule treating $a$ or $b$ as a parameter, but to no avail. Thanks for your help!