List of 2D definite integrals not reducible to products of 1D integrals

I'm writing numerical integration routines for 2D surface integrals. To test it, I'm looking for a list of definite integrals which have analytic forms. I need

• Integrals in polar coordinates over the unit disk
• Integrals over the unit square
• Integrals over triangles and convex hulls
• Integrals over all $\mathbb{R}^{2}$ and the half-plane

I already have many tests for integrals which can be reduced to products of 1D integrals, so I was hoping to find more that are not reducible in this way. I've check Abramowitz & Stegun, the CRC Standard Mathematical Tables & Formulae, and googled around but haven't found enough examples to really challenge the routine and as such I was hoping that the good people of math.se had some truly enjoyable 2D surface integrals they could show me!

Thanks!

Here's what I've found thus far:

• $\int_{2}^{3} \int_{1}^{\infty} x^{-y} \mathrm{d}x \mathrm{d}y = \ln(2)$
• $\int_{0}^{1} \int_{0}^{1} \frac{x}{\sqrt{x^2 + y^2}} \, \mathrm{d}x \mathrm{d}y = \frac{1}{2}(\ln(\sqrt{2}+1) + \sqrt{2} -1)$
• $\int_{0}^{1} \int_{0}^{1} \frac{1}{1-xy} \, \mathrm{d}x \mathrm{d}y = \frac{\pi^2}{6}$

References are also appreciated.