$$\int_{r_{min}}^{\infty} \frac{dr}{r^2 \sqrt{1-\frac{V(r)}{E_c}-\frac{p^2}{r^2}}} $$ where $r_{min}$ is the root of the denominator.
$$V(r)=\frac{Z_1 Z_2 q^2}{4 \pi \epsilon_0 r} \Phi(r)$$
where $a_U$, $q$, $\epsilon_0$, $a_0$, $Z_1$, $Z_2$, $p$ are constant.
I tried some naive solutions, but the problem is that the thing to integrate (let's call it $Y$) approaches infinity when $r$ approaches $r_{min}$. So mathematically, this integral is supposed to converge (or is it?), but numerically it's ill-posed.
In the naive approach, I find $r_{min}$ numerically, but this gives a very high $Y$ at the beginning of the integration (that should be $\pm \infty$ if $r_{min}$ was exact), and then I try to integrate with very small trapezoids, but it seems this approach is fundamentally flawed.
So... I should probably transform this integral, changing variables and stuff, but I don't know where to begin...
Is there a known method to compute this ?
I've googled "exp-sinh quadrature" thanks to user14717 comment.
I tried
boost::math::quadrature::exp_sinhand this works flawlessly, thanks a lot!