An elementary function is a function that can be represented by a finite number of exponentials, logarithms, nth roots, and constants through composition. Clearly, an non-elementary function that is not elementary.
There are plenty of elementary functions such that integrating said function results in a non-elementary functions. Some quick ones that come to mind are $e^{x^2}$, $x^x$, and $\frac{1}{\ln{x}}$.
My question:
Is it possible to find a non-elementary function $f(x)$ such that $\int f(x)dx$ is an elementary function?
My intuition tells me no, but he's been wrong plenty of times before.
The derivative of an elementary function is always an elementary function. Thus, if $\int f(x)dx$ is elementary, $f(x)$ must be elementary.