\[\large\displaystyle \int\limits_0^{\infty} \frac{\log^2(1 - e^{-x})x^5}{e^x - 1} \, dx \]

Let \(I\) denotes the above integral which can be expressed the form of

\(\displaystyle I = A\pi^B\zeta^C(D) - E\zeta(F)\zeta(G) + H\pi^J\zeta(K)\)

Evaluate \(A+B+C+D+E+F+G+H+J+K\)

\(\bullet A,B,....,K\) all in positive integers.

\(\blacksquare\) This is a part of Hot Integrals

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