# 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.

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