In Gravitation by Misner et al. 1973 the authors state that (calculus related to the Schwarzschild metric page 678) :
$$2\int_{0}^{\frac{1}{r_{0}}}\frac{du}{\sqrt{\frac{r_{0}-r_{s}}{r_{0}^{3}}-u^{2}\left(1-u r_{s}\right)}}= 4\sqrt{\frac{r_{0}}{q}}\left(F\left(\frac{\pi}{2}, k\right)-F\left(\sigma_{0}, k\right)\right)-\pi$$ with : $$ \begin{cases} F\left(\sigma, k\right) &= \int_{0}^{\sigma}\frac{dx}{\sqrt{1-k^{2}\sin^{2}x}}\\ q &= \sqrt{r_{0}^{2}+2r_{0}r_{s}-3r_{s}^{2}} \\ k &= \sqrt{\frac{q-r_{0}+3r_{s}}{2q}}\\ \sigma_{0} &= \sin^{-1}\left(\sqrt{\frac{q-r_{0}+r_{s}}{q-r_{0}+3r_{s}}}\right)\\ \end{cases}$$ Neither Maple nor Mathematica seem to be able to find this form.
How to demonstrate it ?