For $\alpha \geq 0$ the transformation $x \mapsto \log(x) + \alpha \log(1-x)$ maps the unit intervall to the real line (in fact for $\alpha = 0$ the transformation is not surjective). For $\alpha=1$ this is the logit transformation, which has a well-known inverse. It is also not difficult to find an inverse for $\alpha = 0.5$ and I would guess along these lines for some more $\alpha = 1/N$, $N$ positive natural number. But I do not see a closed form for an arbitrary $\alpha > 0$.
Do you see one, or can you show there is none?