The introduction of polar coordinates is often motivated (among others) with the possibility of calculating definite integrals of functions with non-elementary antiderivatives. This is the case with the Gaussian integral $\int_{-\infty}^{\infty}e^{-x^2}dx$ which can be easily calculated by computing the corresponding double integral in two ways (one using Fubini's theorem and the other by using polar coordinates).
I am looking for a similar integral that would motivate the introduction of spherical coordinates.
More specifically what I want is
-A definite integral such that the antiderivative is non-elementary.
-For pedagogical reasons it is preferable that the integrand is a quite elementary function
-Can be calculated by using the spherical coordinates and a triple integral (perhaps in the same way as the Gaussian integral)
-Can not be calculated by more elementary methods, specifically the method using polar coordinates fails.