This is a question NOT on how to actually solve the problem but more about a concern in what one needs to know before making a sort of substitution.
Last day I tried to solve this integral: $\int \frac{1}{\sqrt[3]{\tan x}}dx$. Now this integral can fairly easy be found in terms of elementary functions. However, I tried with the universal substitution: $t=\tan (\frac{x}{2})$ which gives me:
$\sin x=\frac{2t}{1+t^2}$ $\cos x=\frac{1-t^2}{1+t^2}$ and further $\tan x=\frac{2t}{1-t^2}$. This didn't really simplify the problem but I plugged in this into Wolfram Alpha which gives a solution in terms of non-elemtary functions(hypergeometric functions). Totally different from the actual solution. I should probably say that i have no knowledge in non-elementary functions.
So where did I go wrong and how can I avoid falling into this trap?
