What are some possible ways to write a program that, following the rules of differentiation, to derive, implicitly, some function built from f(x) and g(x), and end up with f(g(x)) as a result?
Basically, is there a chain rule (and before you ask, I do know what u-substitution is, and that is not what I am looking for) for breaking up
$$\int f(g(x))dx$$
on any closed interval, into any combination of simple integral terms of f(x) or g(x), such as: $\int f(x)dx$, or $\iint g(x)dx$ or any of the derivatives of the two functions, multiplied together?
Even if the answer isn't a simple one, like integrating:
$$\int e^{-x^2}dx$$
using $\int {e^{-x}}dx=-e^{-x} + C$, or $\int {x^2}dx={x^3}/3 + C$, I still think it would be nice to hear a suggestion.