# Complex integration?

Calculus Level 5

$\large \int_0^1 \ln(1-e^x) \, dx$

If we consider the branch of the complex logarithm to be $$\ln(-x) = \ln x + i \pi$$, then the complex integral above is equal to $\text{Li}_a (e^{-b}) + \dfrac cd + f \cdot i\pi - \zeta (g),$ where $$a,b,c,d,f,g$$ are positive integers with $$c,d$$ coprime, find $$a+b+c+d+f+g$$.

Notations:

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

• $$i = \sqrt{-1}$$.

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

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