TITLE: Is there a name for the class of functions which are infinitely integrable in elementary functions?
QUESTION [16 upvotes]: Is there a name for the class of functions which are infinitely integrable in elementary functions, that is whose consecutive integrals also elementary not depending on how much times we took the integral?

REPLY [2 votes]: No, unfortunately there is not name for these functions. In general, the integral of an non-fractional elementary function will result in another elementary function. Polynomials are the name for $ax^2 + bx + c$ sort of things with this being a three term polynomial. In general, I think the term you want is a non-fractional elementary function. Really, this all depends on what you define as elementary. If you include natural logs, and exponents then you would use that as a descriptive term. In general, functions are classes according to their form and basic characteristics, not according to how they react within an integral or derivative.