Just to reiterate: we need to know what you have in mind by an elementary function.
The algebraic approach is to take say polynomials, $\exp(x)$ and perhaps $\ln(x)$ and then to say a function is elementary if it can be written in terms of these using operations of linear combinations, multiplication, composition. Then there are elementary functions with no antiderivative. So the obvious thing to do is to extend the definition of elementary function to include these antiderivatives. The theorem is that this doesn't work.
There are algorithms which are implemented in computer algebra systems which given an elementary function will either find an antiderivative or will prove that no elementary function is an antiderivative.
Response This answers the question the OP asked with the definition of elementary that OP intended. This appears to have caused offence so I wish to state my position.
It is a fact of life that in the classroom and in computer algebra systems that functions are described by formulae. This leads to many problems and the (now defunct) symbolic algebra newsgroup was inundated with complaints from people who did not appreciate the limitations of this approach.
There are several interesting discussions we could have on the issues that arise from this but I don't think this would be welcome on this site.