Reflect on Euler

Calculus Level 5

If

\[\int\limits_0^1 \frac{(\ln x)^2}{x-1}\left[ x^{-1/3} - x^{-2/3} \right]\,\mathrm dx\]

can be expressed in the form \(\dfrac{A\pi^n}{B\sqrt{C}}\), where \(A,B,C\) and \(n\) are positive integers with \(A\) and \(B\) coprime as well as \(C\) squarefree, then find \(A+B+C+n\).

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