I am trying to calculate the following integral. When changing it to polar, then it becomes very ugly. Can one please at least show that it exists or not? I will also be very thankful for a plain evaluation! Hints are always nice of course! $$\iint_{\mathbb{R}^2}\frac{xy}{x^2-xy+1}\exp\left(-2x^2+6xy-6y^2\right)\, dx\, dy$$
Many thanks :-)