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

• To: mathgroup at smc.vnet.net
• Subject: [mg87784] Re: If Integrate returns no result, can we conclude that no closed-form
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Wed, 16 Apr 2008 22:33:26 -0400 (EDT)
• References: <fu4lpc$sk9$1@smc.vnet.net>

Szabolcs_Horvát <szhorvat at gmail.com> 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."
>
> http://reference.wolfram.com/mathematica/tutorial/IntegralsThatCanAndCann
> otBeDone.html
>
> How precise is this?  Can one rely on this information?

I suppose that depends on the definition of "essentially".  ;-)

> Is it really
> true that if Mathematica cannot integrate an expression made up of
> elementary functions, then no closed-form result exists?

No. Consider, for example,

In[9]:= Integrate[D[x Sin[x^ArcSin[x]], x], x]

Out[9]= Integrate[x^(1 + ArcSin[x])*Cos[x^ArcSin[x]]*(ArcSin[x]/x +
          Log[x]/Sqrt[1 - x^2]) + Sin[x^ArcSin[x]], x]

which was done in Versionn 6.0.2 under Windows XP.

> 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).)

People who work for WRI can answer better than I, but if I understand
correctly, Mathematica's implementation of the Risch algorithm is
incomplete. Furthermore, I suspect that there is no current CAS having a
complete implementation of it.

David

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