Evaluating $\int \tan^{1/3}(\theta) d \theta$ I know $\int \tan^{1/3}\theta  d \theta$ is a non integrable but wolfram alpha does it easily. And can anyone explain how a fuction is non integrable? My argument is that there has to be a function which represents the area under a graph. It's just that we do not know it. Please shed a bit light on it. Also, how can we solve $\int \tan^{1/3}\theta  d \theta$?

I will just leave here the exact solution.\begin{align}
\int (\tan x)^\frac13\,dx=&\frac14 \bigg[-2\sqrt{3}  \tan^{-1}\left(\sqrt{3}-2(\tan x)^\frac13 \right)-2\sqrt{3}\tan^{-1}\left(\sqrt{3}+2(\tan x)^\frac13  \right)\\
&-2\log\left(\tan^\frac23x+1\right)+\log\left(\tan^\frac23x -\sqrt{3}(\tan x)^\frac13+1   \right)\\
&\hspace{5mm}+\log\left(\tan^\frac23x+\sqrt{3}(\tan x)^\frac13+1      \right)    \bigg].
\end{align}
