TITLE: A possible closed form?
QUESTION [5 upvotes]: Does it have a closed form?
$$\int _{0}^{\infty }\!{\frac {x\cos \left( x \right) -\sin \left( x
 \right) }{{x} \left( {{\rm e}^{x}}-1 \right) }}{dx}$$
EDIT: no need for answer, I just found the closed form. Thanks!

REPLY [2 votes]: Is this what you have found?
$$\int _{0}^{\infty }\!{\frac {x\cos \left( x \right) -\sin \left( x
 \right) }{{x} \left( {{\rm e}^{x}}-1 \right) }}{dx} = \frac{\pi}{2}+\arg\left(\Gamma\left(i\right)\right)-\Re\left(\psi_0\left(i\right)\right).$$
Here $\arg$ is the complex argument, $\Re$ is the real part of a complex number, $\Gamma$ is the gamma function, $\psi_0$ is the digamma function, $i$ is the imaginary unit and $\pi$ is also a famous constant.