Possibly the strangest calculus problem?

Calculus Level 5

\[\begin{equation} \sum_{n=1}^\infty\int_0^1 \frac{x^{n-1}\ln^2(x)\ln(1-x)}{n^2} \,dx=-\frac{\pi^{\large a}}{b} \end{equation}\]

Find \(\,\large a+b\,\) for the following problem

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