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Topic: An independent integration test suite
Replies: 5   Last Post: May 5, 2013 12:29 PM

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clicliclic@freenet.de

Posts: 983
Registered: 4/26/08
Re: An independent integration test suite
Posted: May 4, 2013 4:11 PM
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Waldek Hebisch schrieb:
>
> clicliclic@freenet.de wrote:

> >
> > Waldek Hebisch schrieb:

> > >
> > > BTW: Core integrator normally only deals with algebraics,
> > > exponential and logarithms, sometimes with tangent (when
> > > the integrand is a rational function of integration variable
> > > and a single tangent). What users see is the result of
> > > a postprocessor. I will probably modify postprocessor
> > > to restore asin-s and acos-es.
> > >

> >
> > I suspect this would not help in all situations: it would not convert
> > the FriCAS term SQRT(x/(x+1))*SQRT(x+1) in example 81 back to my (it is
> > not Derive's) term (x+1)*SQRT(1/(x+1))*SQRT(x/(x+1)), or would it?

>
> Of course asin-s and acos-es are separate from roots. However,
> 'sqrt(1/(x+1))' is indirectly part of user input (as part
> of derivative of 'asin') so restoring user expressions involves
> using 'sqrt(1/(x+1))' too.
>

> > Since the ASIN's in an integrand get improperly converted to (or are
> > improperly interpreted as) ATAN's (perhaps the rational derivative of
> > the latter is preferred) you might consider forestalling this branch-cut
> > violating act by preprocessing them like this:
> >
> > ASIN(z) <- 2*ATAN(z/(1+SQRT(1-z^2)))
> >
> > This is a relation from the Wolfram functions site that holds for
> > arbitrary complex z (assuming the MMA choices of branch cuts). It is
> > also valid in Derive.

>
> This formula may have advantages, thanks. However, most complex
> processing happens in Risch algorithm, so I need to look at
> what consequence it has for Risch. As I wrote atans get
> converted to logs anyway. For logs the factor of two before
> formula means that we are effectively taking log of a square
> root. If that root can be obtained for free, then it is great.
> But if some root which otherwise would get squared survives,
> then it means much harder job for Risch. So it may be
> better to restore asins and acoses from logs, than to
> use different transformation to atans and restore atans.
>


Such a restoration should be able to correct many antiderivatives, but I
also fear that it may be no more than a kludge which can be fooled. Do
you consider this a general solution to problem of branch-cut memory
loss in the FriCAS integrator?

Martin.

PS: With respect to your results at

<http://axiom-wiki.newsynthesis.org/SandboxFriCASTimofeev>,

I think that (#e^(x/2))^2 in the antiderivative for Timofeev's example
62 should be automatically simplified to #e^x.



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