Integrating $\int dx \int dy (x-y)^2 xy \exp(-a(x-y)^2)$

Question

Any useful change of variable possible to make the integration easier ? $$ \int dx \int dy (x-y)^2 xy \exp(-a(x-y)^2) $$

Answer 1

Use a rotation:

$$u = \frac{x-y}{\sqrt{2}}$$ $$v = \frac{x+y}{\sqrt{2}}$$

to get the integral

$$\iint_{\Bbb{R}^2} u^2(v^2-u^2)\exp(-2au^2)\:du\:dv$$

which diverges because of the $v$.