$\large \int_{0}^{1} \dfrac{\ln( x)}{1-x^2} \, dx$

The above integral is equal to $\dfrac{-a\pi^b}{c}$ for positive integers $a,b,c$, for positive integers $a,b$ and $c$ with $a$ and $c$ coprime. Evaluate $(a+b+c)^2$.

**Note**

You are given that $\zeta(2) = \dfrac{\pi^2}{6}$.