Log and exponential integral

Calculus Level 5

\[ \large \int_0^\infty \dfrac{x^3}{e^x - 1} \ln(e^x - 1) \, dx = \pi^A \zeta(B) + C \zeta(D) \]

If \(A,B,C\) and \(D\) are positive integers such that the equation above holds true, find \(A+B+C+D\).

Notation: \(\zeta(\cdot) \) denotes the Riemann Zeta function.

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