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\begin{align}
&\color{#f00}{%
4\int_{0}^{\infty}{\sinh^{2}\pars{x/8} \over x\pars{\expo{x} - 1}}\,\dd x} =
4\int_{0}^{\infty}{x \over \expo{x} - 1}\,
{\sinh\pars{x/8} \over x}\,{\sinh\pars{x/8} \over x}\,\dd x
\\[4mm] = &\
4\int_{0}^{\infty}{x \over \expo{x} - 1}\,\half\int_{-1/8}^{1/8}\expo{kx}\,\dd k
\,\half\int_{-1/8}^{1/8}\expo{qx}\,\dd q\,\dd x =
\int_{-1/8}^{1/8}\int_{-1/8}^{1/8}\int_{0}^{\infty}
{x\expo{-\pars{1 - k - q}x} \over 1 - \expo{-x}}\,\dd x\,\dd k\,\dd q
\\[4mm] = &
\int_{-1/8}^{1/8}\,\,\sum_{n = 0}^{\infty}\,\,\int_{-1/8}^{1/8}\int_{0}^{\infty}
x\expo{-\pars{1 - k - q + n}x}\,\,\,\,\dd x\,\dd k\,\dd q =
\int_{-1/8}^{1/8}\,\,\sum_{n = 0}^{\infty}\,\,\int_{-1/8}^{1/8}
{1 \over \pars{1 - k - q + n}^{\,2}}\,\dd k\,\dd q
\\[4mm] = &\
{1 \over 4}\int_{-1/8}^{1/8}\,\,\sum_{n = 0}^{\infty}
{1 \over \pars{n + 7/8 - q}\pars{n + 9/8 - q}}\,\dd q
\\[4mm] = &\
{1 \over 4}\int_{-1/8}^{1/8}\pars{-4}
\bracks{\Psi\pars{{7 \over 8} - q} - \Psi\pars{{9 \over 8} - q}}\,\dd q =
\left.\vphantom{\huge A^{A}}
\ln\pars{\Gamma\pars{7/8 - q} \over \Gamma\pars{9/8 - q}}
\right\vert_{\ -1/8}^{\ 1/8}
\\[4mm] = &\
\ln\pars{{\Gamma\pars{3/4} \over \Gamma\pars{1}}\,
{\Gamma\pars{5/4} \over \Gamma\pars{1}}} =
\ln\pars{\Gamma\pars{3 \over 4}\,{1 \over 4}\,\Gamma\pars{1 \over 4}} =
\ln\pars{{1 \over 4}\,{\pi \over \sin\pars{\pi/4}}} =
\ln\pars{2^{-3/2}\,\,\pi}
\\[4mm] = &\
\color{#f00}{\ln\pars{\pi} - {3 \over 2}\,\ln\pars{2}}
\end{align}