Tricky Integral Problem with tan and sec function

Question

Can someone help me evaluate:$$\int \frac{(\sec x)^{2}}{(1+\tan x)^{2}}dx$$ Is it possible for a hint so that I can proceed? I tried changing sec into $ 1 +\tan x $ but did not reach far.

Answer 1

Let $u = \tan x$. Then $du = \sec^2 x\, dx$. Thus

$$\int \frac{\sec^2 x}{(1 + \tan x)^2}\, dx = \int \frac{du}{(1 + u)^2} = \cdots$$

Answer 2

How about modifying this integration into 1/(sinx+cosx)^2 and let sinx+cosx=sqrt2 ×sin (x+pi/4)?