Integrate :- $\int dx/(\sin(x) + a\sec(x))^2$

Question

Please help me in evaluating this integral

$$ \int \frac{1}{(\sin(x) + a \sec(x))^2}\,dx $$ I tried by converting $\sec(x)$ to $\cos(x)$ and by solving it became more complicated so guys please guide me further.

Answer 1

Hint: $$\dfrac1{(\sin x+a\sec x)^2}=\dfrac1{2(\sin x\cos x+a)^2}+\dfrac{\cos2x}{2(\sin x\cos x+a)^2}$$

The second part is elementary.

$$\dfrac1{(\sin x\cos x+a)^2}=\dfrac{\sec^2x(1+\tan^2x)}{(\tan x+a\tan^2x+a)^2}$$

Choose $\tan x=u$