How to evaluate the following integral? $\int\frac1{1+\sqrt{\tan x}}\mathrm dx.$

Question

Evaluate the following integral: $$\int\dfrac1{1+\sqrt{\tan x}}\mathrm dx.$$

I know this question has a solution, but I haven't the slightest idea how to do it.

Answer 1

$t=\sqrt{\tan(x)}$ then we get $x=\arctan(t^2)$ and $dx=\frac{2tdt}{1+t^4}dt$ and we get the integral $\int\frac{2tdt}{(1+t)(1+t^4)}$