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Calculus Level 5

\[ \large \int_0^1 \dfrac{ (\ln x)^2 \; \text{Li}_3 (x)}{1-x} \, dx = (\zeta(b))^a - \zeta (c) \]

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

\[ \]

Notations:

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