A simple integral related to the zeta function

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Proposition 1. We have

$${\int }_0^{\mathrm{\infty }}\frac{x^3}{(x^4+1)(e^{2\pi x}-1)}\mathrm{d}x=\frac{\gamma }{2}+\frac{\pi }{4}\frac{\sin \pi \sqrt{2}}{\cosh \pi \sqrt{2}-\cos \pi \sqrt{2}}-\frac{1}{8}\sum _{{\omega }^4=-1}{H}_{\omega },$$

where $H_s=\gamma +{\psi }_0(1+s)$ is the analytic extension of the harmonic number.