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

*To*: mathgroup at smc.vnet.net*Subject*: [mg87810] Re: [mg87784] Re: If Integrate returns no result, can we conclude that no closed-form*From*: "David W. Cantrell" <DWCantrell at sigmaxi.net>*Date*: Thu, 17 Apr 2008 06:59:39 -0400 (EDT)*References*: <fu4lpc$sk9$1@smc.vnet.net> <200804170233.WAA21093@smc.vnet.net> <5E0C371E-10BB-43A7-BE86-C2A06FE723D6@mimuw.edu.pl>

Andrzej Kozlowski wrote: > On 17 Apr 2008, at 11:33, David W.Cantrell wrote: >>> 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/IntegralsThatCanAndCannotBeDone.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. > This is not an "elementary function" (as far as the Risch algorithm > is concerned). > > The class of elementary functions consists of rational functions, > exponentials, logarithms, trigonometric, inverse trigonometric, > hyperbolic, and inverse hyperbolic functions, the solutions of a > polynomial equation whose coefficients are elementary functions) and > any finite nesting (composition) of elementary functions. It does > not include the power of an elementary function to another > elementary function so your example actually shows the opposite of > what you seem to claim, namely that Mathematica has correctly > concluded (by using the Risch algorithm) that the integral cannot be > expressed in terms of elementary functions. Are you claiming that, for the purpose of the Risch algorithm, Exp[Log[x] ArcSin[x]] is not considered to be an elementary function? I find that hard to believe! David

**References**:**Re: If Integrate returns no result, can we conclude that no closed-form***From:*"David W.Cantrell" <DWCantrell@sigmaxi.net>