[Animats]

You mean integration? That's much harder, but has been done automatically. Symbolic differentiation is easy, because you can just keep blindly applying a set of rewrite rules until none of them apply. That process converges on a unique result. Symbolic integration doesn't converge in that way. More strategy is required, and you're not guaranteed a closed form solution. Mathematica has a good symbolic integrator.

[cperciva]

Symbolic integration doesn't converge in that way. More strategy is required, and you're not guaranteed a closed form solution.

However, if a closed-form solution exists which can be expressed in terms of the operations + - * / exp log, then it is guaranteed to be found.

[tome]

Reverse mode AD is another algorithm for calculating derivatives

    https://en.wikipedia.org/wiki/Automatic_differentiation#Reverse_accumulation