Polylogs in Integrals

Calculus Level 5

01Li3(x)ln(1x)xdx \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

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

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

Find A+B+C+D+E+F+G+HA+B+C+D+E+F+G+H.

Notations:

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

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

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