A little bit of both. I don't think WA uses the full Risch algo, but even then, enough pattern matching rules https://rulebasedintegration.org/ https://rulebase

A little bit of both. I don't think WA uses the full Risch algo, but even then, enough pattern matching rules https://rulebasedintegration.org/ https://rulebasedintegration.org/ beats it in perf. Differentiation under the integral sign only works for certain well behaved functions and isn't easy to automate since you now need to figure out where to parametrize and you don't have good structure theorems to help you. IMO contour integration is a more powerful and easier to intuit technique.