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Calculus Level 5

\[\large \displaystyle\int _{ 0 }^{ 1 }{ \ln { \left( 1-\cos { x } \right)\, dx } } \] The above integral has the form:

\[\dfrac { i }{ a } -\dfrac { i{ \pi }^{ b } }{ c } +\ln { \left( \dfrac { d }{ ({ e }^{ fi }-g)^{ h } } \right) } +\ln { \left( j-\cos { k } \right) } +l\cdot i\cdot \text{Li}_m(e^{ni})\]

for positive integers \(a,b,c,d,f,g,h,j,k,l,m\) and \(n\).

Find \(a+b+c+d+f+g+h+j+k+l+m+n\).

Notations:

  • \(i=\sqrt{-1}\)

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

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