I have heard that most elementary functions don't have elementary antiderivatives. Is there a precise meaning to the previous sentence, and if so, may I see a paper where the precise version of that statement is proven?
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1See Risch algorithm and Liouville's theorem. – Lucian Sep 03 '14 at 05:11
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To make it precise, you'd presumably have to specify a particular probability measure on the elementary functions. They really don't have any "natural" probability measure, so this choice would have to be quite arbitrary. – Robert Israel Sep 03 '14 at 05:19
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Or maybe you could sample functions from some particular source (say the pages of a book or journal). But in some cases, most of the functions found there might be rational, and therefore would have elementary antiderivatives. – Robert Israel Sep 03 '14 at 05:22
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The answer depends on what is an "elementary" function. The boundary between a list of so called "elementary" functions and the infinity of other functions is rather arbitrary. And even more between a list of so called "special" functions and the others.
A trivial approach to tackle the problem would be to make a list of "elementary" functions and for each of them to express the antiderivative when it is possible. Then, compare the number of antiderivatives which are included in the list of elementary functions and the number of those which are not in the list or which have not been found.
The similar question is raised for the antiderivatives of "special" functions. This is a high level problem which involve the Liouville's theorem and relationships with the Gallois theory. Without going so far, a review paper for general public pubished on Scribd relates to elementary functions which antiderivatives are not elementary, but are special functions and even more, are at the origin of the definition of new special functions : http://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
JJacquelin
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