While I was working on this question, I've found that
$$
I=\int_0^\infty\frac{x\cos(x)-\sin(x)}{x\left({e^x}-1\right)}\,dx = \frac{\pi}2+\arg\left(\Gamma(i)\right)-\Re\left(\psi_0(i)\right),
$$
where $\arg$ is the complex argument, $\Re$ is the real part of a complex number, $\Gamma$ is the gamma function, $\psi_0$ is the digamma function.
How could we prove this? Are there any more simple closed-form?
A numerical approximation:
$$
I \approx -0.3962906410900101751594101405188072631361627457\dots
$$
Answer
$$\begin{align}
\int_0^{\infty} \frac{x\cos x-\sin x}{x(e^x-1)}dx=\int_0^{\infty} \frac{(x\cos x-\sin x)e^{-x}}{x(1-e^{-x})}dx
\\=\int_0^{\infty} \frac{x\cos x-\sin x}{x}\sum_{n=1}^{\infty}e^{-xn}\,\,dx
\\=\sum_{n=1}^{\infty}\left(\Re\int_0^{\infty}e^{-xn}e^{ix}dx-\tan^{-1}\left(\frac1{n}\right)\right)
\\=\sum_{n=1}^{\infty}\left(\Re\frac{1}{n-i}-\tan^{-1}\left(\frac1{n}\right)\right)
\\=\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}-\tan^{-1}\left(\frac1{n}\right)\right)
\\=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\tan^{-1}\left(\frac1{n}\right)-\frac{1}{n(n^2+1)}\right)
\\=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{(2m+1)n^{2m+1}}-\Re\sum_{n=1}^{\infty}\frac{i}{n(n+i)}
\\=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\zeta(2n+1)}{2n+1}-\Re{H_{i}}
\\=\Im\sum_{n=2}^{\infty} \frac{(-1)^n\zeta(n)}{n}i^n-\Re(H_{i}-\frac1{i})
\\=\Im\left(i\gamma+\ln(i)+\log(\Gamma(i))\right)-\Re H_{i-1}
\\=\gamma+\frac{\pi}{2}+\operatorname{arg}\Gamma(i)-\Re\left(\gamma+\psi(i)\right)
\\=\frac{\pi}{2}+\operatorname{arg}\Gamma(i)-\Re\psi(i).
\end{align}$$
Where I used $\,\,\,\,\,\displaystyle \sum_{n=2}^{\infty}\frac{(-1)^n\zeta(n)}{n}x^n=\gamma x+\log\Gamma(x+1)$
which follows from the logarithm of the Weierstrass-form Gamma function,
and $\,\,\,\,\,\displaystyle \int_0^{\infty} \frac{\sin x}{x}e^{-nx}dx=\tan^{-1}\left(\frac1{n}\right)\,\,\,\,$ is a standard exercise in differentiation under the integral sign.
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