In some situations, there is a huge performance advantage to symbolic integration. The symbolic integration is expensive, but once it's done, you can evaluate integrals of different parameterisations of your function in a handful of cycles. With numerical integration, you have to spend however many thousands of cycles it takes for every parameterisation.

If you're evaluating integrals as part of the objective function in an optimisation over a multidimensional space, then you're going to be doing a lot of optimisations, and that saving can add up.