TITLE: Closing up the elementary functions under integration
QUESTION [27 upvotes]: The elementary real-valued functions are not closed under integration.  (Elementary function has a precise definition --  see Risch algorithm in Wikipedia).  This means there are elementary functions whose integrals are not elementary.  So we can construct a larger class of functions by adjoining all the integrals of elementary functions.  You can repeat this process indefinitely.  If I understand things correctly, the set of functions that is the countable closure of this process is closed under integration.  Does any finite iteration of the process achieve closure under integration?
My guess is no.  Has anyone thought about this?

REPLY [3 votes]: In a differential Galois field, integration is the antiderivative operation and not the infinite summation, but I will still call it integration.  
If we start out with simple elementary functions, say rational functions with integer coefficients, then there will be many functions that do not have integrals. If the original set of functions is finite, then only a finite number of functions need to be added.  The process can then be repeated at the next level.  The only catch is that at every stage one has to deal with constants (think of the roots of the denominators). Most functions force the addition of several integrals.  The number of functions added grows geometrically with the levels, so I would expect there is no easy way to define a countable closure.
Are there initial sets for which the process stops? Yes, polynomials for example.  Are there initial sets for which it goes on forever?  Yes.