Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$
$$\frac{x-\sin x}{x^2}=\sum_{n\geq 1}\frac{(-1)^{n+1}}{(2n+1)!}x^{2n-1},$$ and since: $$ \int_{0}^{+\infty}\frac{x^{2n-1}}{e^x-1}\,dx = (2n-1)!\cdot \zeta(2n),$$ we have: $$\begin{eqnarray*} &&\int_{0}^{+\infty}\frac{x-\sin x}{x^2(e^x-1)}\,dx = \sum_{n\geq 1}\frac{(-1)^{n+1}}{2n(2n+1)}\zeta(2n)\\&=&\color{red}{\sum_{n\geq 1}\left(-1+n\arctan\frac{1}{n}+\frac{1}{2}\,\log\left(1+\frac{1}{n^2}\right)\right)}\\&=&\color{blue}{\log\sqrt{\frac{\sinh \pi}{\pi}}+\sum_{n\geq 1}\left(-1+n\arctan\frac{1}{n}\right)}.\tag{1}\end{eqnarray*} $$ Combining this identity with the robjonh's answer to another question, we finally get: $$\color{purple}{\int_{0}^{+\infty}\frac{x-\sin x}{x^2(e^x-1)}\,dx=\frac{1}{2}+\frac{5\pi}{24}-\log\sqrt{2\pi}+\frac{1}{4\pi}\operatorname{Li}_2(e^{-2\pi})}.\tag{2}$$
On the other hand, the identity claimed by user111187, $$ \int_{0}^{+\infty}\frac{x-\sin x}{x(e^x-1)} = \gamma+\Im\log\Gamma(1+i)\tag{3} $$ follows from the integral representation for the $\log\Gamma$ function and for the Euler-Mascheroni constant. By considering the Weierstrass product for the $\Gamma$ function, $$\Gamma(z+1) = e^{-\gamma z}\prod_{n\geq 1}\left(1+\frac{z}{n}\right)^{-1}e^{\frac{z}{n}}$$ we have: $$ \log\Gamma(z+1) = -\gamma z + \sum_{n\geq 1}\left(\frac{z}{n}-\log\left(1+\frac{z}{n}\right)\right)$$ so: $$ \int_{0}^{+\infty}\frac{x-\sin x}{x(e^x-1)}\,dx = \sum_{n\geq 1}\left(\frac{1}{n}-\arctan\frac{1}{n}\right).\tag{4}$$
$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\dsc}[1]{\displaystyle{\color{red}{#1}}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\Li}[1]{\,{\rm Li}_{#1}} \newcommand{\norm}[1]{\left\vert\left\vert\, #1\,\right\vert\right\vert} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\int_{0}^{\infty}{x - \sin\pars{x} \over \pars{\expo{x} - 1}x^{2}}\,\dd x: \ {\large ?}}$.
\begin{align}&\color{#66f}{\large% \int_{0}^{\infty}{x - \sin\pars{x} \over \pars{\expo{x} - 1}x^{2}}\,\dd x} =\int_{0}^{\infty}{1 \over \pars{\expo{x} - 1}x}\,{x - \sin\pars{x} \over x}\,\dd x \\[5mm]&=\int_{0}^{\infty}{1 \over \pars{\expo{x} - 1}x}\,\ \overbrace{\half\int_{-1}^{1}\pars{1 + \ic k x - \expo{\ic k x}}\,\dd k} ^{\dsc{x - \sin\pars{x} \over x}}\ \,\dd x\ =\ \half\int_{-1}^{1}\int_{0}^{\infty}{1 + \ic k x - \expo{\ic k x} \over \pars{\expo{x} - 1}x}\,\dd x\,\dd k \\[5mm]&=\half\int_{-1}^{1}\int_{0}^{\infty} {\expo{-x} + \ic k x\expo{-x} - \expo{-\pars{1 - \ic k}x} \over 1 - \expo{-x}}\ \overbrace{\int_{0}^{\infty}\expo{-xt}\,\dd t}^{\dsc{1 \over x}}\ \dd x\,\dd k \\[5mm]&=\half \int_{-1}^{1}\int_{0}^{\infty}\sum_{n\ =\ 0}^{\infty}\ \int_{0}^{\infty} \bracks{\expo{-\pars{n + 1 + t}x}\ +\ \ic k x\expo{-\pars{n + 1 + t}x}\ -\ \expo{-\pars{n + 1 + t - \ic k}x}\,\,} \,\dd x\,\dd t\,\dd k \\[5mm]&=\half \int_{-1}^{1}\int_{0}^{\infty}\sum_{n\ =\ 0}^{\infty}\ \bracks{{1 \over n + 1 + t}\ +\ {\ic k \over \pars{n + 1 + t}^{2}}\ -\ {1 \over n + 1 + t - \ic k}}\,\dd t\,\dd k \\[5mm]&=\half \int_{-1}^{1}\int_{0}^{\infty}\sum_{n\ =\ 0}^{\infty}\ \bracks{{\ic k \over \pars{n + 1 + t}^{2}} -{\ic k \over \pars{n + 1 + t}\pars{n + 1 + t - \ic k}}}\,\dd t\,\dd k \\[5mm]&=\half\int_{-1}^{1}\int_{0}^{\infty} \bracks{\Psi\pars{1 + t -\ic k} - \Psi\pars{1 + t}}\,\dd t\,\dd k \end{align} where $\ds{\Psi}$ is the Digamma Function.
Then, \begin{align}&\color{#66f}{\large% \int_{0}^{\infty}{x - \sin\pars{x} \over \pars{\expo{x} - 1}x^{2}}\,\dd x} =\left.\half\int_{-1}^{1} \ln\pars{\Gamma\pars{1 + t - \ic k} \over \Gamma\pars{1 + t}} \right\vert_{\, t\ =\ 0}^{\, t\ \to\ \infty}\,\dd k \\[5mm]&=-\,\half\int_{-1}^{1}\ln\pars{\Gamma\pars{1 - \ic k}}\,\dd k =\dsc{-\,\Re\int_{0}^{1}\ln\pars{\Gamma\pars{1 - \ic k}}\,\dd k}\tag{1} \end{align}
However, \begin{align}&\dsc{-\,\Re\int_{0}^{1}\ln\pars{\Gamma\pars{1 - \ic k}}\,\dd k} =-\,\Re\int_{0}^{1}\ln\pars{-\ic k\Gamma\pars{-\ic k}}\,\dd k \\[5mm]&=-\int_{0}^{1}\ln\pars{k}\,\dd k -\Re\int_{0}^{1}\ln\pars{\Gamma\pars{-\ic k}}\,\dd k =1-\Re\int_{0}^{1} \ln\pars{\pi \over \Gamma\pars{1 + \ic k}\sin\pars{\pi\bracks{-\ic k}}}\,\dd k \\[5mm]&=1 - \ln\pars{\pi} + \Re\int_{0}^{1}\ln\pars{ \Gamma\pars{1 + \ic k}}\,\dd k +\Re\int_{0}^{1}\ln\pars{-\ic\sinh\pars{\pi k}}\,\dd k \\[5mm]&=1 - \ln\pars{\pi} +\dsc{\Re\int_{0}^{1}\ln\pars{ \Gamma\pars{1 - \ic k}}\,\dd k} +\int_{0}^{1}\ln\pars{\sinh\pars{\pi k}}\,\dd k \\[1cm]&\imp\quad \dsc{-\,\Re\int_{0}^{1}\ln\pars{\Gamma\pars{1 - \ic k}}\,\dd k} =\half - \half\,\ln\pars{\pi} +\half\int_{0}^{1}\ln\pars{\sinh\pars{\pi k}}\,\dd k \\[5mm]&=\half - \half\,\ln\pars{\pi} +\half\int_{0}^{1}\bracks{\pi k + \ln\pars{1 - \expo{-2\pi k}} - \ln\pars{2}} \,\dd k \\[5mm]&=\half - \half\,\ln\pars{2\pi} + {1 \over 4}\,\pi -\half\sum_{n\ =\ 1}{1 \over n}\int_{0}^{1}\expo{-2n\pi k}\,\dd k \\[5mm]&=\half - \half\,\ln\pars{2\pi} + {1 \over 4}\,\pi -\half\sum_{n\ =\ 1}{1 \over n}{\expo{-2n\pi} - 1 \over -2n\pi} \\[5mm]&=\half - \half\,\ln\pars{2\pi} + {1 \over 4}\,\pi +{1 \over 4\pi}\ \overbrace{\sum_{n\ =\ 1}{\pars{\expo{-2\pi}}^{n} \over n^{2}}} ^{\dsc{\Li{2}\pars{\expo{-2\pi}}}}\ -{1 \over 4\pi}\, \overbrace{\sum_{n\ =\ 1}{1 \over n^{2}}}^{\dsc{\pi^{2} \over 6}} \\[5mm]&=\half - \half\,\ln\pars{2\pi} + {5 \over 24}\,\pi +{1 \over 4\pi}\,\Li{2}\pars{\expo{-2\pi}} \end{align}
Replacing in expression $\pars{1}$: \begin{align}&\color{#66f}{\large% \int_{0}^{\infty}{x - \sin\pars{x} \over \pars{\expo{x} - 1}x^{2}}\,\dd x} =\color{#66f}{\large\half - \half\,\ln\pars{2\pi} + {5 \over 24}\,\pi +{1 \over 4\pi}\,\Li{2}\pars{\expo{-2\pi}}} \\[5mm]&\approx {\tt 0.2357} \end{align}
This is another solution just for reference.
The laplace transform of $\displaystyle\frac{x-\sin{x}}{x^2}$ is given by \begin{align} \mathcal{L}_s\left(\frac{x-\sin{x}}{x^2}\right) &=\int^s_\infty\int^t_\infty\frac{1}{u^2}-\frac{1}{1+u^2}\ du\ dt\tag1\\ &=\int^s_\infty-\frac{1}{t}-\arctan{t}+\frac{\pi}{2}\ dt\\ &=-\ln{s}+\frac{1}{2}\ln(1+s^2)-s\arctan{s}+\frac{\pi}{2}s-1\\ &=\frac{1}{2}\ln\left(1+\frac{1}{s^2}\right)+s\arctan\left(\frac{1}{s}\right)-1 \end{align} Thus \begin{align} \int^\infty_0\frac{x-\sin{x}}{x^2(e^x-1)}dx &=\sum^\infty_{n=1}\mathcal{L}_n\left(\frac{x-\sin{x}}{x^2}\right)\tag2\\ &=\frac{1}{2}\sum^\infty_{n=1}\ln\left(1+\frac{1}{n^2}\right)+\sum^\infty_{n=1}\left(n\arctan\left(\frac{1}{n}\right)-1\right)\\ &=\frac{1}{2}\ln\left.\frac{\pi z}{\pi}\prod^\infty_{n=1}\left(1+\frac{z^2}{n^2}\right)\right|_{z=1}+\sum^\infty_{n=1}\sum^\infty_{k=1}\frac{(-1)^k}{(2k+1)n^{2k}}\tag{3}\\ &=\frac{1}{2}\ln\left(\frac{\sinh(\pi)}{\pi}\right)+\sum^\infty_{k=1}\frac{(-1)^k\zeta(2k)}{(2k+1)}\tag4\\ &=\frac{1}{2}\ln\left(\frac{\sinh(\pi)}{\pi}\right)+\frac{1}{2i}\int^{i}_0\left(1-\pi z\cot(\pi z)\right)\ dz\tag5\\ &=\frac{1}{2}\ln\left(\frac{\sinh(\pi)}{\pi}\right)+\frac{1}{2}-\frac{1}{8\pi}\int^{\exp(-2\pi)}_1\frac{\ln{u}(1+u)}{u(1-u)}du\tag6\\ &=\frac{1}{2}\ln\left(\frac{\sinh(\pi)}{\pi}\right)+\frac{1}{2}-\frac{1}{8\pi}\left[2\mathrm{Li}_2(1-u)+\frac{\ln^2{u}}{2}\right]^{\exp(-2\pi)}_1\tag7\\ &=\color{#E2062C}{\frac{1}{2}-\frac{\pi}{4}+\frac{1}{2}\ln\left(\frac{\sinh(\pi)}{\pi}\right)-\frac{1}{4\pi}\mathrm{Li}_2\left(1-e^{-2\pi}\right)} \end{align} Explanation:
$(1)$: Differentiated under the integral twice.
$(2)$: Expanded $(e^{x}-1)^{-1}$.
$(3)$: Expanded $\arctan\left(n^{-1}\right)$.
$(4)$: Recognised the Weierstrass product for $\sinh$, summed in $n$.
$(5)$: Used the fact that $\displaystyle\pi z\cot(\pi z)=1-2\sum^\infty_{k=1}\zeta(2k)z^{2k}$.
$(6)$: Substitued $u=e^{2\pi iz}$.
$(7)$: $\displaystyle \frac{\ln{u}(1+u)}{u(1-u)}=\frac{2\ln{u}}{1-u}+\frac{\ln{u}}{u}$.