Polylogs in Integrals

Calculus Level 5

\[ \large \int_0^1 \dfrac{ \text{Li}_3(x) \ln(1-x)} x \, dx \]

The value of the integral above can be expressed in the form of

\[ \dfrac AB \pi^C \zeta (D) - \dfrac EF \pi^G \zeta (H) \; , \]

where \(A,B,C,D,E,F,G,H\) are non-negative integers, where \(\gcd(A,B) = \gcd(E,F) = 1 \).

Find \(A+B+C+D+E+F+G+H\).

Notations:

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

  • \(\zeta(\cdot) \) denotes the Riemann zeta function.

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