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>