Really Simple Integral?

Calculus Level 4

\[ \large\displaystyle\int_{0}^{\frac{\pi}{2}}\ln^{2}(\sin(x))\mathrm dx=\frac{{\pi}^{3}}{a}+\frac{\pi}{b}\ln^{2}{(c)} \] If the equation above is true for constants \(a,b,c\), find \(a+b+c\).

This is part of Ordered Disorder.

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