What does \(\pi\) have to do with \(\ln\) integrals ?

Calculus Level 5

\[\int_0^1 \frac{\ln x \left(\ln(1-x)\right)^2}{x} dx = \frac{-a \pi^b}{c}, \]

where \(a,b,c\) are positive integers and \(\gcd(a,c)=1.\) Find \(a+b+c.\)

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