integration – Closed form of beta distribution CDF when $alpha + beta = 1$

Is there a closed form of the regularized incomplete Beta function $I_x(alpha, beta)$ when $alpha + beta = 1$?

A nice closed form based on arcsin exists for the case where $alpha + beta = 1/2$ (see here). Is there a similar one for the more general case?

My motivation is that I’d like to find, given $p in (0, 1)$, the $alpha$ such that the $Pr(X > 0.5) = p$ where $X sim Beta(alpha, 1- alpha)$. I would like to do this efficiently for any $p in (0, 1)$ – which should be easy with the closed form of the regularized Beta function.