# 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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