It isn't what it looks!

Calculus Level 5

02πx2(ln2sinx2)2dx=AπCB \displaystyle \int_0^{2\pi} x^2\left( \ln \left| 2 \sin\dfrac{x}{2} \right| \right)^{2} \, dx= \dfrac{A\pi^C}{B}

If the equation above holds true for integers A,BA,B and CC with A,BA,B coprime positive integers, find A+B+CA+B+C.

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