\begin{equation} \int_0^{2 \pi}\int_0^{2 \pi} \frac{\cos\left(u\right) + \cos\left(v\right)-\cos\left(u\right)\cos\left(v\right)}{\,\sqrt{\,\left\{\vphantom{\LARGE A}2 \cos\left(u\right)\left[\cos\left(v\right)-1\right]-2\cos\left(v\right)+3 \right\}^{3}\,}\,}\,{\rm d}u\,{\rm d}v \end{equation}
I know the result is 1, because the above integral represent the linking number of 2 closed curves which in the pair diagram cross twice. However, how to compute it analitically?
