I am trying to solve an integral from the following function (int) and set of assumptions (as):
int = Exp[(-a)*(r^2 + r1^2 + 2*r*r1*(Cos[f]*Cos[f1] +
Cos[t - t1]*Sin[f]*Sin[f1]))]*Sin[f]*r^2;
as = Assumptions -> {Element[{a, r, r1, f, f1, t, t1, a},
Reals], a >= 0, r >= 0, r1 >= 0, t >= 0, t <= Pi,
t >= 0, t1 <= Pi, f >= 0, f1 <= 2*Pi};
The integral is:
Integrate[int, {t, 0, 2*Pi}, {f, 0, Pi}, {r, 0, s}, as]
which seems hopeless to obtain by brute force. I have no problem in obtaining the integral numerically, but it is very costly as I need to compute it many times. Do you have ideas on how to solve it (at least w.r.t. 2 variables would be great and then treat the third variable numerically)? I could obtain analytical expressions w.r.t. one of each of the three variables. E.g.,
intt = Integrate[int, {t, 0, 2*Pi}, as]
yields:
intt = (2*Pi*r^2*BesselI[0, 2*r*r1*a*Sin[f]*Sin[f1]]*Sin[f])/
E^(a*(r^2 + r1^2 + 2*r*r1*Cos[f]*Cos[f1]))
But then, I get stuck if I try to integrate w.r.t. the remaining variables (including assumptions above does not help):
inttr = Integrate[intt, {r, 0, s}]
inttf = Integrate[intt, {f, 0, Pi}, as]