Let $z$ be a complex number. Let $f(z)$ be an elementary function but not a polynomial. Let its integral $F(z)$ be impossible to express in elementary functions. If we define $F(z)$ as $\int$ from $A$ to $z$ (in a straight path from $A$ to $z$), how do we know if there is a finite complex $A$? What if there are multiple values where $F(z)$ = $0$; What $A$ to choose? How to compute $A$? What $A$ makes the computation of the integral the easiest? Is it the $A$ closest to $z$?
If $f$ is an entire function that is not a polynomial and $F$ is an antiderivative of it, the little Picard theorem says $F$ takes on every complex value with at most one exception. That exception could be $0$, in which case $F = \exp(G)$ for some entire function $G$. Offhand I couldn't think of an example of this where $f$ is elementary but $F$ is not.
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