Simple looking integrals can be very nasty

Calculus Level 5

I=0π/2x log2(sin(x)) dx 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 II can be represented as Li4(AB)CπDE+logF(G)H+πMlogN(P)Q { \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 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 A,B,C,E,F,G,H,M,N,P,Q are positive integers not necessarily distinct,C,E C,E are co-prime to each other, A,BA,B are co-prime to each other , G,PG,P are not perfect power of any integer (that it is not a perfect square, cube etc.)

2)Lis(z)=k=1zkks \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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