Numerical results on the integral expression for the semiclassical S matrix are compared with exact quantum results for a multidimensional problem. The collision of a rigid rotor with an atom is treated. The integral method proves to be easy to ·apply. Within its range of maximum validity (no sign changes in the pre-exponential factor of the semiclassical wavefunction) the agreement was typically within 20%. When sign changes occurred, the agreement was about a factor of 2 or better. Conditions affecting sign changes are described.