<s>[INST] integral $\int_0^{\pi} \left( \frac{\pi}{2} - x \right) \frac{\tan x}{x} \, {\rm d}x$ Evaluate , if possible in a closed form, the integral:
$$\int_0^{\pi} \left( \frac{\pi}{2} - x \right) \frac{\tan x}{x} \, {\rm d}x$$
Basically, I have not done that much. I broke the integral 
\begin{align*}
\int_{0}^{\pi} \left ( \frac{\pi}{2}-x \right ) \frac{\tan x}{x} \, {\rm d}x  &= \int_{0}^{\pi/2} \left ( \frac{\pi}{2} - x \right ) \frac{\tan x}{x} \, {\rm d}x  + \int_{\pi/2}^{\pi} \left ( \frac{\pi}{2} - x \right ) \frac{\tan x}{x} \, {\rm d}x\\ 
 &\!\!\!\!\!\!\overset{u=\pi/2-x}{=\! =\! =\! =\! =\! =\!} \int_{0}^{\pi/2} \frac{u \cot u}{\frac{\pi}{2}-u} \, {\rm d}u + \int_{-\pi/2}^{0} \frac{u \cot u}{\frac{\pi}{2}-u} \, {\rm d}u\\ 
 &= \int_{-\pi/2}^{\pi/2} \frac{u \cot u}{\frac{\pi}{2}-u} \, {\rm d}u\\ 
 &\approx 2.13897
\end{align*}
I have no idea how to evaluate this. I was thinking of IBP and then some kind of Fourier , but I cannot get it to work. Any ideas?
 [/INST] Too long for a comment. We have the following representations:
\begin{align*}
\int_{0}^{\pi} \left(\frac{\pi}{2} - x\right) \frac{\tan x}{x} \, dx
&= \frac{\pi^2}{4} + \frac{\pi}{2}\int_{0}^{1} \psi\left(\frac{1+x}{2}\right) \sin(2\pi x) \, dx \\
&= \frac{\pi^2}{4} - \pi \int_{0}^{\infty} \frac{dt}{(1+t^2)(e^{2\pi t} + 1)}.
\end{align*}
At this moment I have no good idea of dealing with these, though I am posting this in hope of providing an alternative start for others as well as for a self record.
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