All Roads Lead To \(\pi\)

Calculus Level 3

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

If the value of the integral above equals \(\dfrac{\pi ^{a!}}{a+1}\), where \(a\) is a positive integer, find \(a\).