My question arises from this post: Gauss's Law with Moving Charges
I have tried everything I can think of to solve the integral, and I have absolutely no clue as to how it was solved. I typed it into Wolfram Alpha and attempted to get a step-by-step solution but the computation time exceeded and it wasn't able to show me a step-by-step. I think my issue comes from the denominator being raised to $3\over 2$. How was this integral solved?
\begin{align} \oint \vec{E} \cdot d\vec{a} &= \oint \frac{k\, q\, \vec{r} \cdot \hat{r}}{r^3}\, \frac{1-v^2}{(1-v^2\, \sin^2\theta)^{3/2}}\, r^2\, \sin \theta\, d\theta\, d\phi \\ &= k\, q\, \int_0^{2\pi} \int_0^\pi \frac{1-v^2}{(1-v^2\, \sin^2\theta)^{3/2}}\, \sin \theta\, d\theta\, d\phi\\ &= 2\pi\, k\, q\, \int_0^\pi \frac{1-v^2}{(1-v^2\, \sin^2\theta)^{3/2}}\, \sin \theta\, d\theta \\ &= 4\pi\, k\, q~, \end{align}