A Cool Integral in Disguise

Calculus Level 5

\[ \large \displaystyle \int_0^\infty \dfrac{x^7 (e^{3x} - e^x)}{(e^x -1)^4} \, dx = \dfrac{n! \pi^n}{m} \]

If the above integral is true for positive integers \( n\) and \(m\), what is the value of \(n + m\)?

You may want to look up the value of \( \zeta(2n) \) for a certain positive integer \(n\).

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