Yes, but symbolic differentiation is straight forward. There are simple rules to apply to differentiate any expression: the chain rule, the product rule, the q

Yes, but symbolic differentiation is straight forward. There are simple rules to apply to differentiate any expression: the chain rule, the product rule, the quotient rule, derivatives of known single argument functions, etc. Integration is a completely different ball game. Given an arbitrary expression, most likely it's not symbolically integrable, i.e. it's integral can't be expressed as a combination of known functions. That's why there are lots of functions defined as "the integral of blah." Even given an expression that's known to be symbolically integrable, it's quite the puzzle to figure out what the integral is. There's no straight forward algorithm, as with differentiation. It boils down to "try integration by parts; try this; try that; maybe do some clever algebraic manipulation to get it into a form where you can try one of those things; look it up in a book of integrals to see if it's there, or something similar is there; if all else fails, guess and check."