I don't see any problem with the result.
In[4]:= Cos[ArcTan[x, y]]
(* Out[4]= x/Sqrt[x^2 + y^2]*)
So the result looks to be correct (do it by substitution t=x^2 + y^2, dt=2*x, so it reduces to Integrate[Exp[-j*Sqrt[t]]/Sqrt[t], t] up to multiplicative constants.
Maybe you had in mind to use Cos[ArcTan[y/x]]?
