I like integrals

Calculus Level 5

$\displaystyle\int _{ -2 }^{ -1 }{ \dfrac { \ln { { (x }^{ 2 }-x) } }{ x } } \, dx =\frac { 1 }{ A } ({ \pi }^{ B }-C{ \ln { ^{ D }{ F } } })+{ Li }_{ 2 }(-G)$

where $$A,B,C,D,F,G$$ are positive integers with $$F$$ minimized

find $$A+B+C+D+F+G$$

Notation: $${ \text{Li} }_{ n }(a)$$ denotes the polylogarithm function, $${ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \frac { { a }^{ k } }{ { k }^{ n } } }.$$

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