<s>[INST] An analytical solution of a tricky integral Can anyone propose a method (some methods) to determine the following indefinite integral?
$$I=\int\frac{\textrm{d}x}{\sqrt[3]{\sqrt{1+x^{2}}-x}}.$$
I think an analytical solution should be possible...
 [/INST] Using the substitution $x = \sinh t$, we have $dx = \cosh t dt$.
\begin{align}
I &= \int\frac{dx}{\sqrt[3]{\sqrt{x^2 + 1}-x}}\\
&= \int\frac{\cosh t}{\sqrt[3]{\cosh t - \sinh t}}\,dt\\
&= \int e^{t/3}\cosh t \,dt\\
&= \frac{1}{2}\int\left(e^{4t/3} + e^{-2t/3}\right)\,dt\\
&= \frac{1}{2}\left(\frac{3}{4}e^{4t/3} - \frac{3}{2}e^{-2t/3}\right) + C
\end{align}
Recalling our substitution, which implies that $x = \operatorname{arsinh}t$, and using that 
$$\exp(\beta\operatorname{arsinh}x) = \left(\sqrt{x^2+1}+x\right)^\beta$$
we obtain that
$$I = \frac{3}{8}\left(\sqrt{x^2+1}+x\right)^{4/3} - \frac{3}{4}\left(\sqrt{x^2+1}+x\right)^{-2/3} + C$$
</s>