# Who's up to the challenge? 45

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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