Mutidimensional integration with cosine rule in exponent I have been attempting to find a way to simplify the following multi-dimensional integration form:

$\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\int\limits_{0}^{\pi}
 e^{\left(-a-b+\sqrt{a^2+b^2-2ab\cos{x}}\right)} \left(ba\sqrt{a^2+b^2-2ab\cos{x}}\right)dx \; da \; db$

I have used a variety of numerical methods which all provide a number given the infinities in the integration domains are changed for a finite number. When trying by hand to simplify it, the exponential function seems to be the sticking point for me. Would approximating it as a series help?
This particular part of the mathematics is slowing down my code quite dramatically; hence I was wondering if there is any mathematical trickery to simplify it?

If you look at the expression, you'll see that it doesn't converge; there's an exponential function with a positive exponent integrated to $\infty$. No matter what integration method you use (including making a series expansion), you will get $\infty$.
