Integration

\int \frac{dx}{\sin^2x+\tan^2x}
= \int\frac{dx}{(1-\cos^2x)+\frac{1-\cos^2 x}{\cos^2 x}}
= \int \frac{\cos^2x\,dx}{(1-\cos^2x)(1+\cos^2 x)}
=\frac{1}{2}\int \left ( \frac{1}{1-\cos^2x}-\frac{1}{1+\cos^2 x}\right ) dx
=\frac{1}{2}\int \csc^2 x\,dx-\frac{1}{2}\int \frac{\sec^2 x\,dx}{\sec^2x+1}
=\frac{-\cot x}{2}-\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+C