I let Mathematica run overnight the following code
Integrate[(r^2 Cos[A r Cos[x]] Exp[-r^2/2])/(a^2 Cos[x]^2 +
b^2 Sin[x]^2), {r, 0, Infinity}, {x, 0, Pi/2}]
but it was unable to compute the closed form in that time. However, Mathematica also didnt give up. Do you think Mathematica can do it?
