A simple integral related to the zeta function

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Proposition 1. We have

\begin{equation*} \int_{0}^{\infty} \frac{x^3}{(x^4 + 1)(e^{2\pi x} - 1)} \, \mathrm{d}x = \frac{\gamma}{2} + \frac{\pi}{4} \frac{\sin \pi\sqrt{2}}{\cosh \pi\sqrt{2} - \cos \pi\sqrt{2}} - \frac{1}{8} \sum_{\omega^{4} = -1} H_{\omega}, \end{equation*}

where $H_s = \gamma + \psi_{0}(1+s)$ is the analytic extension of the harmonic number.