To integrate or not to integrate, that is the question

$\large{\displaystyle \int_{0}^{\infty} \frac{x^{3}e^{-x}}{1-e^{-x}}}$

$\large{\displaystyle \int_{0}^{\infty} x^3 \displaystyle \sum_{r=1}^{\infty} e^{-rx}}$

$\large{\displaystyle \sum_{r=1}^{\infty} \displaystyle \int_{0}^{\infty} x^3 e^{-rx} dx}$

put $rx=t$

$\large{= \Gamma (4)\displaystyle \sum_{r=1}^{\infty} \frac{1}{r^4} }$ $\left (\because \displaystyle \int_{0}^{\infty} t^{n} e^{-t} dt=\Gamma (n+1) \right )$

$\large{= \Gamma (4) \zeta (4)}$

$\large{=\boxed{\frac{\pi^{4}}{15}}}$