Expanded concept of elementary function? After searching about why $\int e^{x^2}$ is not an elementary function, I was disappointed that I should understand about Galois theory, but then I started to think about a concept that treats elementary functions, anti-derivative of elementary functions, and operations (+-*/, composition) of those functions, which includes non-elementary functions like $\int e^{x^2}$, $\int \frac{sin(x)}x$, or $\int log(log(x))$.
I do believe that some people in past thought about that, but I simply don't know how to search.
What's the name of the type of functions that can be derived by differentiate, integrate, and composite of elementary functions? Plus, just out of curiosity, is there any function that's infinitely differentiable but not in the type of functions I asked?
 A: You might be interested in a more extreme type of non-elementary function. A function $y$ of $x$ is said to be "transcendentally transcendental" on an interval $(a,b)$ if $P(x,y,y', y'',...,y^{n})$ is not identically zero on $(a,b)$ for every positive integer $n$ and every nonzero polynomial $P$ of $n+2$ variables. In other words, $y$ doesn't satisfy any algebraic differential equation, including non-linear algebraic differential equations. None of the elementary transcendental functions (trigonometric, exponential, logarithmic) are transcendentally transcendental on any open interval, and most of the higher functions in mathematical physics aren't either (elliptic functions, Bessel functions, etc.). However, in 1887 Hölder proved that the gamma function is transcendentally transcendental, which incidentally gives a naturally occurring example of an infinitely differentiable function that is far more non-elementary than what you were asking about. (I'm not sure, but I think Hölder was also the first to formulate the property of being "transcendentally transcendental".) A nice survey paper of this topic is:
Lee Albert Rubel, "A survey of transcendentally transcendental functions", American Mathematically Monthly 96 (1989), 777-788.
Many of the older papers on this topic, including Hölder's original paper and some papers by E. H. Moore (in a 1897 paper, E. H. Moore introduced the name "transcendentally transcendental") and some papers by J. F. Ritt (1923, 1926) are in Math. Annalen, and thus are freely available on the internet. There's also a 1902 paper by Edmond Maillet in Bulletin de la Societe Mathematique de France (Vol. 30, pp. 195-201) that is freely available on the internet.
A: The point of Liouville's theorem is that functions that have elementary integrals must have a special form, no matter how you define elementary function. See How can you prove that a function has no closed form integral?. My answer there contains a list of readable references.
A: The function class that contains all elementary functions as defined by Liouville and all antiderivatives of the elementary functions are the Liouvillian functions. The Liouvillian functions are the closure under indefinite integration of the set of the elementary functions. This function classes are defined e.g. in section 1 of Davenport, J. H.: What Might "Understand a Function" Mean. In: Kauers, M.; Kerber, M., Miner, R.;  Windsteiger, W.: Towards Mechanized Mathematical Assistants. Springer, Berlin/Heidelberg, 2007, page 55-65.
The Liouvillian functions are only a small part of the closed-form functions. Not all named Special functions are Liouvillian functions. E.g. the power series or the generalized hypergeometric functions are infinitely differentiable but not all in the Liouvillian functions.
