The sum below numerically (to 13th digit at least) is the same as $\ln 2$. So there should be a way to prove it analytically, but I haven't succeeded. Any suggestions?
$$\frac\pi2+8\sum_{n=1}^\infty\cfrac{(-1)^n\tanh\frac{n\pi}2+((-1)^n-1)\coth n\pi}{n\pi}\int_0^1\frac{\cos n\pi t}{1+t^2}dt =^? \ln 2$$
The integral in the sum is fourier coefficient of $\frac1{1+t^2}$ on $[0,1]$. It can be taken in terms of the exponential integrals $E_i(z)$, but this didn't lead me to anything. My last thought was to change somehow the order of summation-integration but I can't remember the series with hyperbolic functions in it.
Thanks for help.