Continuing the series of horrible integrals, my instructor gave me exercise to solve next indefinite integral:
$$\int \frac{dx}{\sqrt{\tan x}} $$
Seems simple and short, but wolframalpha gives me totally horrible answer.
Is there any way to simplify this integral or any hints on solving it? Maybe some trigonometric formulas?
Let $u=\sqrt{\tan x}$ ,
Then $x=\tan^{-1}u^2$
$dx=\dfrac{2u}{u^4+1}~du$
$\therefore\int\dfrac{dx}{\sqrt{\tan x}}=\int\dfrac{2}{u^4+1}~du$
The only key point is how to evaluate $\int\dfrac{du}{u^4+1}$ .
You can factorize $u^4+1$ and partial fraction decomposition as usual (as foolish as WolframAlpha), or getting the smarter approach e.g. in Evaluating $\int \frac{1}{{x^4+1}} dx$.