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Re: If Integrate returns no result, can we conclude that no

On Apr 16, 5:52 am, Szabolcs Horv=E1t <szhor... at> wrote:
> 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."
> 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).)

No, just the fact that Integrate cannot do an integral doesn't mean no
closed form in elementary functions exists. Here's one example that
Integrate cannot currently do but can be done in closed form (in terms
of elementary functions):

Integrate[(1 - x^2 - 5*x^4 + 9*x^5 - 5*x^6 + 5*x^9)/(Sqrt[-1 + x +
x^5]*(-1 + x - x^2 + x^5)), x]

Integrate does not currently have a full implementation of the
algebraic case of the Risch algorithm, for instance, although we still
do pretty well using substitutions and elliptic integration code for
such inputs.

Wolfram Research

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