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If Integrate returns no result, can we conclude that no closed-form

• To: mathgroup at smc.vnet.net
• Subject: [mg87759] If Integrate returns no result, can we conclude that no closed-form
• From: Szabolcs Horvát <szhorvat at gmail.com>
• Date: Wed, 16 Apr 2008 06:52:44 -0400 (EDT)
• Organization: University of Bergen

The documentation says:

"In the most convenient cases, integrals can be done purely in terms of 
elementary functions such as exponentials, logarithms and trigonometric 
functions. In fact, if you give an integrand that involves only such 
elementary functions, then one of the important capabilities of 
Integrate is that if the corresponding integral can be expressed in 
terms of elementary functions, then Integrate will essentially always 
succeed in finding it."

http://reference.wolfram.com/mathematica/tutorial/IntegralsThatCanAndCannotBeDone.html

How precise is this?  Can one rely on this information?  Is it really 
true that if Mathematica cannot integrate an expression made up of 
elementary functions, then no closed-form result exists?

Szabolcs

(P.S. I do not know how Integrate works.  I heard that CASs use a 
so-called "Risch-alogrithm", but there is relatively little information 
about this on the web (except in academic papers, most of which expect 
the reader to be familiar with the topic).)

• Follow-Ups:
  • Re: If Integrate returns no result, can we conclude that no closed-form
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>