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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>