I have been reading as to why there are elementary functions that have non-elementary antiderivatives and I have come to the conclusion that our notion of "elementary" functions is somewhat arbitrary, stemming mostly from the types of problems that arise in the world. Are there any other candidate elementary functions that would expand our ability to take antiderivatives?
In my attempt at formality:
Let $E$ be the set of all "elementary operations" (exp,sin,cos,logarithms,polynomials,powers,constants,etc...), and let $F$ be the set of all finite compositions of $E$. If I am correct, $F$ should be our entire set of elementary functions. Is there a way to make $E$ as small as possible? Basically form it into a basis that spans $F$, where our operation is a composition.
One last part, for any function $x \in F$, let $D(x)$ be the derivative of $x$ and let $I(x)$ be the integral of $x$. It is fairly obvious that $D(F) \subseteq F$, and that $F \subset I(F)$. But how much bigger is $I(F)$ than $F$? Can we throw more "elementary functions" into $E$ and define $F$ in the same way, such that $I(F) = F$? Is it always true that for any set of functions $F$ that $F \subset I(F)$? I haven't read any sort of theory about this sort of idea and so it feels like I'm discovering a new type of math.
Note: To add some structure to this question, I think it would be best to have the universal set of functions be real (or perhaps complex) functions that are infinitely differentiable, but perhaps this is too limiting.
Here are a couple ideas that might help.
One options is to think about so-called analytic functions. A function $f(x)$ is analytic on a set (usually an open interval in $\mathbb{R}$) if is equal to a convergent power series on that interval. Analytic functions have all the have nice algebraic properties that if you add, multiply, differentiate, integrate, or compose analytic functions you get another analytic function (on perhaps a smaller domain.) These are good candidates for your elementary functions.
Note that differentiable functions in $\mathbb{R}$ and functions in $\mathbb{C}$ are very different. If a function in $\mathbb{R}$ has a derivative, it's second derivative may not exist. For functions in $\mathbb{C}$ is differentiable if and only if it is infinitely differentiable.
Another option to consider for a set of "elementary functions" are measurable functions from the theory of measurable functions and Lebesgue integration. Without really going into the details, the set of Riemann integrable functions over the reals isn't quite enough to handle a lot of important important functions that appear in Fourier analysis for things like signal processing.
If you only want to consider functions that are like the familiar ones from algebra and any finite combinations (addition, multiplication, composition, etc.) I think analytic functions are a natural choice.
On the other hand, measure theory gives that (most) integrable functions can be approximated by piecewise linear functions, so in a sense you can generate all of those integrable functions as the limit of a sequence of piecewise linear functions. Being piecewise linear could be a very solid choice for you set of elementary functions that generate the other "elementary" functions (by generating them as a limits of convergent sequences).
One final alternative it to be MORE restrictive and only consider algebraic functions as elementary (so ONLY algebraic operations, no transcendental functions like $e^x$) and all other functions as "non-elementary" in some sense if they can only be defined by more complicated methods of analysis.