Have you learned Taylor Series? It seems to accomplish what you're attempting. Not to be a cynic, but I'm pretty sure that when trying to apply this to functions with nonelementary inverses you'll end up just integrating to that inverse(ei the nonelementary inverse will be the only function with the derivative in question). It does have application in finding inverses of functions with elementary inverses, but I, for one, would prefer to stay as far away from integrating (3x^{2}+2x+1)^{1} as possible.
