# How to evaluate $\int \frac{\mathrm dx}{\sqrt[3]{\tan\,x}}$?

Please show me the steps of the following integration. I got an answer in Wolfram, but I need steps..

$$\int \frac{\mathrm dx}{\sqrt[3]{\tan\,x}}$$

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@draks here it is. And WA do not know closed form for this integral –  Norbert Jul 31 '12 at 11:02
Well, I'm sorry its not arctan, its just tan –  Anubis Jul 31 '12 at 11:05
This is the link –  Anubis Jul 31 '12 at 11:08
@draks sorry, I've typed it wrong. It's now corrected. Can you evaluate now? –  Anubis Jul 31 '12 at 11:10
Aha, so please edit your question accordingly!!! and click on "show steps" in your linked W|A page... –  draks ... Jul 31 '12 at 11:10

We try the substitution $t^3 = \tan^2 x$. Therefore, $3t^2 dt = 2 \tan x \sec^2 x dx$, giving us $\frac{dx}{\sqrt{t}} = \frac{3 dt}{2(1+t^3)}$.
Thus, we will only evaluate $\int \frac{3 dt}{1+t^3}$, divide by $2$ and substitute back. Note that $3 = (1-t+t^2) + (2-t)(1+t)$, reducing our integral to $$\int \frac{dt}{1+t} + \int \frac{(2-t)dt}{1-t + t^2}$$ I won't elaborate further, since our integrals are already in standard forms.