Polylogs!

Calculus Level 5

\[\large \int _{ 0 }^{ 1 }{ \dfrac {\text{Li}_2(x^3) }{ x^{ 2 } } \, dx } \]

if the above integral is in the form \[a\ln { \sqrt { b } } -\dfrac { \pi \sqrt { c } }{ d } -\dfrac16 { \pi }^{ f }, \]

where \(a,b,c,d\) and \(f\) are positive integers, 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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