Simple looking integrals can be very nasty

Calculus Level 5

\[ I = \displaystyle \int _{ 0 }^{ \pi /2 }{ x \ { \log }^{ 2 }(\sin(x)) \ \mathrm{d}x } \]

Here's a simple integral for you that has a nasty result.

If \(I\) can be represented as \[ { \text{Li} }_{ 4 }\left(\dfrac { A }{ B } \right)-\dfrac { C{ \pi }^{ D } }{ E } +\dfrac { { \log }^{ F }(G) }{ H } +\dfrac { { \pi }^{ M }{ \log }^{ N }(P) }{ Q } \]

Find \( A+B+C+D+E+F+G+H+M+N+P+Q \)

Details and Assumptions

1)\( A,B,C,E,F,G,H,M,N,P,Q \) are positive integers not necessarily distinct,\( C,E\) are co-prime to each other, \(A,B\) are co-prime to each other , \(G,P\) are not perfect power of any integer (that it is not a perfect square, cube etc.)

2)\( \displaystyle { \text{Li} }_{ s }(z)=\sum _{ k=1 }^{ \infty }{ \frac { { z }^{ k } }{ { k }^{ s } } } \). It is commonly known as Polylogarithm.

I myself am trying to find the closed form of this integral for about a month, and finally found it.
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