A hyperbolic sum

Calculus Level 5

\[ \large \sum_{n=1}^\infty \dfrac{\coth(\pi n) }{n^3} = \dfrac{A\pi^B}C \]

The equation above holds true for positive integers \(A,B\) and \(C\) with \(A,C\) coprime. Find \(A+B+C\).

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