How does one go about evaluating the double integral $$ \int_0^{2\pi} \int_0^{2\pi} \frac{\sin(s) - \sin(t) - \sin(s)\sin(t)}{\big(3 + 2(\sin(s) - \sin(t) - \sin(s)\sin(t))\big)^{3/2}}\,ds\,dt ? $$
It seems like one should be able to evaluate this integral through a combination of trig identities and change of variables. If that is the case, I haven't found the right ones yet.
This integral is the Gauss linking integral for the Hopf link. The linking number of a two-component link with components parameterized by $\gamma_1, \gamma_2 \colon S^1 \to \mathbb{R}^3$ is equal to the integral $$ \frac{1}{4\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{(\gamma_2(s) - \gamma_1(t)) \cdot (\gamma_1'(s) \times \gamma_2'(t)}{\| \gamma_1(s) - \gamma_2(t)\|^3}\,ds\,dt. $$ To arrive at the integral in question, I parameterized the Hopf link by $$ \gamma_1(s) = \begin{bmatrix} \cos(s)\\ \sin(s)\\ 0 \end{bmatrix}, \qquad \gamma_2(t) = \begin{bmatrix} 0\\ \sin(t) - 1\\ \cos(t) \end{bmatrix} $$ and simplified.
I understand that the linking number of the Hopf link parameterized above is $-1$ by combinatorial considerations -- and therefore the integral in question is $-4\pi$ -- but I am curious to see if this integral can be evaluated directly.
Thanks!