Need help in integrating $\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{\cos t-\sin s - \cos t \sin s}{\sqrt{3+2(\cos t - \sin s - \cos t \sin s)}^3}dtds$

This is not a homework problem. I was trying to use the Gaussian linking integral to compute the linking number of the Hopf link and I get stuck at:

$$\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{\cos t-\sin s - \cos t \sin s}{\sqrt{3+2(\cos t - \sin s - \cos t \sin s)}^3}dtds$$

Can anyone suggest some help?

• Calculating it numerically with Maple, I obtain -9.192212570. If the linking number should be integer, then something to adjust. – user64494 Jun 9 '13 at 9:51
• ah yes, I found that I missed a 3 in the exponent of the denominator – wilsonw Jun 9 '13 at 9:56
• In this case (i.e. with the exponent 3) the integral under consideration equals -12.56637061 which is identified as $-4 \pi.$ – user64494 Jun 9 '13 at 10:31
• Thank you; that's my expected answer, as there should be a factor $\dfrac{1}{4\pi}$ in front of the integral. – wilsonw Jun 9 '13 at 11:01
• This integral has been exactly calculated with Maple in MaplePrimes. – user64494 Jul 1 '13 at 18:38