\[ \large \int_0^\pi x^2 \left[ \ln \left( 2\cos \left( \dfrac x2\right) \right) \right]^2 \, dx = \dfrac AB \pi^C \]

The equation above holds true for positive integers \(A,B\) and \(C\) with \(A,B\) coprime. Find \(A+B+C\).

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