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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