In the science paper labelled "Effect of Fermi surface geometry on electron-electron scattering", by Hodges, Smith and Wilkins, there is a following identity:
$ \int_{0}^{ \infty}dx\int_{0}^{ \infty}dz f(z)\left[ 1- f(x) \right]\left[ 1- f(t+z-x)\right] = \frac{1}{2}(\pi ^2 + t^2)\left[ 1- f(t)\right] $
where
$f(x) = \frac{1}{e^x + 1}$
Now, can anyone tell me is there some fancy way to prove it, without the "brute- force" method.
Thanks.