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This I read in a paper:

"The class of integrals that are elementary is very small compared with nonelementary integrals."

What is the precise meaning of this sentence? E.g., does that mean that the former class of functions is meagre (in a suitable functional space) while the latter is not ? Is there a reference for such a subject ?

Thank you

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It is small in the same sense that the set of polynomials solvable by radicals is small. The canonical reference on the subject is probably the late, lamented Manuel Bronstein's book: Symbolic Integration I: Transcendental Functions (Algorithms and Computation in Mathematics) (v. 1) [Hardcover] Otherwise, look up "differential algebra" or "Risch Algorithm".

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  • $\begingroup$ I know all about Risch Algorithm, but I can't find a reference (theorem) concerning the "measure of smallness" of that set. I'll look up Manuel Bronstein's book. $\endgroup$
    – Miguel Ramos
    Commented Oct 20, 2011 at 18:34

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