how do you know when there is no antiderivative written in elementary functions?

Question

I know that there are integrals that cannot be written in this way such as $e^{-x^2}$ but there is a general rule that can be used to recognize these types of integrals?

Answer 1

Theorem: Given $f, g \in C(X)$ with $f \neq 0$ and $g \neq const$. $f(x)e^{ g(x)}$ can be integrated in elementary functions $\iff \exists R ∈ C(X)$ such that $$ R'(X) + g'(X)R(X) = f(X)$$ has a solution in $C(X)$.

$C(X)$ is essentially a field of rational functions.

There is a whole theory behind this, so the proof cannot be given in a single post here. If you want to delve deeper, you need a strong abstract algebra background, differential algebra specifically. Look into Liouville's theorem in particular.