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Calculus Level 5

\[ \large -2 \int_0^1 \dfrac{ (\ln x)^3 \; \text{Li}_4 (x) }{1-x} \, dx = a \; \zeta (b) \]

If the equation above holds true for positive integers \(a\) and \(b\), find \(a+b\).

\[ \]

Notations:

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