It is well-known that taking the double integral of a Bivariate Gaussian Distribution across a rectangle aligned with the axes is very trivial.
However, I am currently considering this double integral on a rotated square. Since now the two variables are related during integration, it seems that computing the outer integral might necessarily involve the multiplication of the Gaussian PDF with the Error Function, which seems to have no analytical solution.
Is there a way to find an analytical solution to this? Any help would be much appreciated!
Note: This is not a homework question, so hints and clues would be gratuitous.