A Cool Integral in Disguise

Consider , $\dfrac{e^{-x}}{1-e^{-x}}$ It can also be written as $\displaystyle \sum _{r=1}^\infty e^{-rx}$ Differentiating twice , we get $\dfrac{(e^{3x} - e^x)}{(e^x -1)^4} = \displaystyle \sum_{r=1}^\infty r^{2}e^{-rx}$ Hence, The integral becomes $\large \displaystyle \int_0^\infty \dfrac{x^7 (e^{3x} - e^x)dx}{(e^x -1)^4} = \displaystyle \sum _{r=1}^{\infty} r^2 \int_0^ \infty x^7 e^{-rx} dx$ Setting $rx$ as $t$ , we get $\displaystyle \sum_{r=1}^{\infty} \dfrac{1}{r^6} \int_0^ \infty t^7 e^{-t} dt \\ = \Gamma(8) \zeta(6) = \dfrac{7! \pi^6}{945} = \dfrac{6!\pi^6}{135}$ So, $n+m = \boxed{\boxed{ 141}}$