Many elementary functions, like $e^{-x^2}$ and $\frac{\sin(x)}{x}$ have antiderivatives that are are nonelementary; is this property generic? That is, does the set of all elementary functions whose antiderivatives are nonelementary form residual (or second category) subset (that is, the complement of a meager or first category subset) of the elementary functions (with some version of the compact-open topology on them, presumably: the elementary functions don't all share a common domain)?
On a related (but still very basic) note, what would be a catch-all term for the collection of all elementary and nonementary functions taken together as one set? "Functions of a real (or complex) variable"?
$\underline{\text{Edit 1}}$: I guess Liouville's Theorem is a partial answer, at least. It appears to yield that the elementary functions whose antiderivatives are elementary are emphatically a meager set, but if I could just get someone who is much more experienced at this game to confirm that for me in simple terms that will be easy for me to understand, I would be most appreciative.
$\underline{\text{Edit 2}}$: In response to an answer from Robert Israel below, I changed the question from asking if the elementary functions whose antiderivatives are nonelementary form an open, dense subset to asking if they form a residual set.
$\underline{\text{Aside}}$: Why isn't Liouville's Theorem part of a standard graduate curriculum for those who want to go on to teach calculus? In what standard (graduate or undergraduate) course would one typically encounter Liouville's Theorem? What book would be a reference for differential algebras and Liouville's Theorem?
Not open, no. Polynomials are dense in the compact-open topology on continuous functions on an interval (or a disk in the complex plane), and they always have elementary antiderivatives.