<s>[INST] Evaluate $\int_0^\infty (u\log(a^2+u^2))/(e^u-1)\, du$ Does the following definite integral have a known "closed form" value?
$$\int_0^\infty\frac{u\log(a^2+u^2)}{e^u-1}~du,$$
or can anyone see a way to integrate it?
 [/INST] I was only able to find the following formula: for $a = 2\pi\alpha$,
$$\begin{align*}
\int_{0}^{\infty} \frac{u \log(4\pi^2\alpha^2+u^2)}{e^u - 1} \; du
&= 2\zeta'(2) + \frac{\pi^2}{3}(1-\gamma) \\
&\quad + \pi^2 \alpha \left(2\alpha \log\alpha - \alpha + 2 - 4 \log\Gamma(\alpha+1)\right) \\
&\quad + 4\pi^2 \int_{0}^{\alpha} \log\Gamma(u+1)\;du.
\end{align*}$$
Indeed, let
$$I(a) = \int_{0}^{\infty} \frac{u \log(a^2+u^2)}{e^u - 1} \; du.$$
Then we have
$$\begin{align*}I(0)
& = 2 \int_{0}^{\infty} \frac{u \log u}{e^u - 1} \; du = 2 \left. \frac{d}{ds}\zeta(s)\Gamma(s) \right|_{s=2} \\
& = 2(\zeta'(2) + \zeta(2)\psi_0(2)) = 2\left(\zeta'(2) + \frac{\pi^2}{6}(1-\gamma)\right).
\end{align*}$$
Also,
$$\begin{align*}
I'(a)
&= \int_{0}^{\infty} \frac{2au}{u^2+a^2} \frac{du}{e^u-1} \\
&= 2a \int_{0}^{\infty} \sum_{n=1}^{\infty} e^{-nu} \left(\int_{0}^{\infty} \sin(ux)e^{-ax}\;dx\right)\;du \\
&= 2a \int_{0}^{\infty} \sum_{n=1}^{\infty} \left(\int_{0}^{\infty} \sin(xu)e^{-nu}\;du\right)e^{-ax}\;dx \\
&= 2a \int_{0}^{\infty} \left(\sum_{n=1}^{\infty} \frac{x}{x^2+n^2}\right)e^{-ax}\;dx \\
&= a \int_{0}^{\infty} \left(\pi\coth(\pi x) - \frac{1}{x}\right)e^{-ax}\;dx
\end{align*}$$
Proceeding,
$$\begin{align*}
I'(a)
&= a \left[ \left(\log\sinh(\pi x)-\log x \right)e^{-ax} \right]_{0}^{\infty} + a^2 \int_{0}^{\infty} \left(\log\sinh(\pi x)-\log x \right)e^{-ax} \; dx \\
&= -a\log\pi + a^2 \int_{0}^{\infty} \left(\pi x + \log\left(1 - e^{-2\pi x}\right)-\log2 + \log a -\log ax \right)e^{-ax} \; dx \\
&= \pi + a\left(\gamma + \log\left(\frac{a}{2\pi}\right)\right)+a^2 \int_{0}^{\infty} e^{-ax}\log\left(1 - e^{-2\pi x}\right) \; dx \\
&= \pi + a\left(\gamma + \log\left(\frac{a}{2\pi}\right)\right) - a^2 \int_{0}^{\infty} \sum_{n=1}^{\infty} \frac{1}{n} e^{-(a+2n\pi)x} \; dx \\
&= \pi + a\left(\gamma + \log\left(\frac{a}{2\pi}\right)\right) - a \sum_{n=1}^{\infty} \frac{\frac{a}{2\pi}}{n\left(n + \frac{a}{2\pi}\right)} \\
&= \pi + a\left(\gamma + \log\left(\frac{a}{2\pi}\right)\right) - a \left(\gamma + \psi_0 \left( \frac{a}{2\pi} +1 \right)\right) \\
&= \pi + a\left(\log\left(\frac{a}{2\pi}\right) - \psi_0 \left( \frac{a}{2\pi} +1 \right)\right).
\end{align*}$$
Thus integrating,
$$\begin{align*}
I(a)
&= I(0) + \int_{0}^{a} I'(t) \; dt \\
&= 2\zeta'(2) + \frac{\pi^2}{3}(1-\gamma) + \pi a + \int_{0}^{a} t \left(\log\left(\frac{t}{2\pi}\right) - \psi_0 \left( \frac{t}{2\pi} +1 \right)\right) \; dt \\
&= 2\zeta'(2) + \frac{\pi^2}{3}(1-\gamma) + 2 \pi^2 \alpha + 4\pi^2 \int_{0}^{\alpha} u \left(\log u - \psi_0 (u+1)\right) \; du \\
&= 2\zeta'(2) + \frac{\pi^2}{3}(1-\gamma) \\
&\quad + \pi^2 \alpha \left(2 \alpha \log \alpha - \alpha + 2 \right) - 4\pi^2 \int_{0}^{\alpha} u \psi_0 (u+1) \; du \\
&= 2\zeta'(2) + \frac{\pi^2}{3}(1-\gamma) \\
&\quad + \pi^2 \alpha \left(2 \alpha \log \alpha - \alpha + 2 \right) - 4\pi^2 \left[  u \log\Gamma(u+1) \right]_{0}^{\alpha} \\
&\quad + 4\pi^2 \int_{0}^{\alpha} \log\Gamma(u+1) \; du,
\end{align*}$$
which gives the desired result. You may verify this formula numerically with the following Mathematica code
a = 8;
NIntegrate[(u Log[4 Pi^2 a^2 + u^2])/(Exp[u] - 1), {u, 0, Infinity}]
2 Zeta'[2] + Pi^2/3 (1 - EulerGamma) + 
 Pi^2 a (2 a Log[a] - a + 2 - 4 LogGamma[1 + a]) + 
 4 Pi^2 NIntegrate[LogGamma[1 + u], {u, 0, a}]
Clear[a];



In fact, we can give a neater form as follows:
$$ \begin{align*}
\int_{0}^{\infty} \frac{u \log(a^2+u^2)}{e^{2\pi u} - 1} \; du
&= \zeta'(-1) + \frac{a}{2}\left(1 - \frac{a}{2}\right) + \frac{a^2}{2}\log a - a \log\Gamma(a+1)\\
&\quad + \int_{0}^{a} \log\Gamma(t+1) \; dt.
\end{align*}$$
</s>