I know this question may sound naïve but why can't we write $\int e^{x^2} dx$ as $\int e^{2x} dx$? The former does not have an antiderivative, while the latter has.
In light of this question, what are sufficient conditions for a function NOT to have an antiderivative. That is, do we need careful examination of a function to say it does not have an antiderivative or is there any way that once you see the function, you can right away say it does not have an antiderivative?