How to take the integral $\int \sqrt{\tan x}\,dx$

Question

Have never really handled that, cause have no clue how to start with it (the ansver seems really long and no hints like $\tan(x/2)$ in the final answer).

$$\int \sqrt{\tan x}\,dx$$

Answer 1

Say $\tan x =t^2$.

Then we have $$\sec^2 x dx=2tdt$$ $$\implies (1+\tan^2 x) dx=2t dt$$ $$\implies (1+ t^4) dx=2t dt$$ $$\implies dx=\frac{2t}{1+ t^4} dt$$

Then the integral becomes $$\int t \cdot \frac{2t}{1+ t^4} dt$$ $$=\int \frac{2t^2}{1+ t^4} dt$$

I hope you can follow up.