Recently, I have been discussing inverses with a tenth grade class and integrals with an eleventh/twelfth grade class, and this has led me to the following wonder:

Wonder. Is there a "reasonable" way to quantify which of finding an inverse and finding an antiderivative is more "difficult"? Meaning, which of these processes is "easier" or which of these processes is more likely to lead to a "pleasant" inverse or antiderivative?

For example, one can use composition to create functions that are easy to invert but difficult to find an antiderivative for; e.g., $x \mapsto e^{x^3}$. (More generally, I am thinking about Liouville's Theorem on when it's possible to express an antiderivative using elementary functions.) On the other hand, if one simply restricted to the set of polynomials in $\mathbb{R}[x]$, then they would all be straightforward to integrate, but finding a pleasant inverse for those of degree five or higher would be almost always impossible. (In this polynomial context, I am thinking about the Abel-Ruffini Theorem.)

Ways to refine the language of this question or pointers to pre-existing explorations of related wonders are all welcome. (Efforts to re-tag will be welcomed, too.)