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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