Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?
Just think of how we find those elementary functions:
We start with the constant functions, which have derivative 0, and the identity function f(x)=x which has derivative 1.
We combine functions by means of addition, subtraction, multiplication, division, composition. For all of those cases we have explicit rules for the derivative.
We define new functions as the integral of other functions (e.g. lnx as integral of 1/x). Obviously when deriving those we get back the function we started with.
We define functions as the inverse of another function. Again, we’ve got an explicit formula for derivatives of inverse functions.
Any function that cannot be defined by a chain of such operations (and also some which can, using the integration rule) we don’t consider elementary.
So basically the reason is in the way we construct elementary functions. In some sense, one could say it is because of what functions we consider elementary.
Indeed, this hold not only for elementary functions; even most non-elementary functions we use are defined through such operations (in particular by integrals).