Who's up to the challenge? 49

Calculus Level 5

\[\large \int _{ 0 }^{ 1 }{ x\text{Li}_2(x^2-1) \, dx } \]

If the integral above can be expressed as \[-\left[ \dfrac { a }{ b } +\dfrac { { \pi }^{ c } }{ d } \right] +\ln { f }, \]

where \(a,b,c,d\) and \(f\) are positive integers, with \(a,b\) coprime, find \(a+b+c+d+f\).

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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