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Topic: An independent integration test suite
Replies: 128   Last Post: Dec 8, 2013 3:21 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The A. F. Timofeev symbolic integration test suite
Posted: Oct 11, 2013 1:50 PM

Albert Rich schrieb:
>
> On Sunday, October 6, 2013 6:44:47 AM UTC-10, clicl...@freenet.de wrote:

> >
> > I always suspected that Derive wants to keep some of its secrets from
> > the competition! And I am not glad to hear that Rubi no longer returns
> > continuous antiderivatives for this type of integrand - but I suppose
> > that's easily put back in. Otherwise I concur absolutely.

>
> Derive does not intentionally keep secret its method of finding
> continuous antiderivatives, and having Rubi generate continuous ones
> is nontrivial. Unfortunately the information required to achieve
> continuity is lost when single-stepping through the integration, so
> Derive is only able to return a discontinuous antiderivative.
>
> Discontinuous antiderivatives often result when the substitution
> u=tan(theta) or u=tan(theta/2) is used to integrate trigonometric
> expressions. The algorithm Derive uses to make such antiderivatives
> continuous involves taking the difference of two limits to find the
> piecewise constant function required to achieve continuity. It is
> described in the 1994 paper I co-authored with D.J. Jeffrey The
> Evaluation of Trigonometric Integrals Avoiding Spurious
> Discontinuities at
> http://www.apmaths.uwo.ca/~djeffrey/Offprints/toms1994.pdf.
> The 1994 Jeffrey paper The Importance of Being Continuous at
> http://www.apmaths.uwo.ca/~djeffrey/Offprints/importan.pdf
> describes how to merge this piecewise constant term with the
> discontinuous inverse tangent term to obtain the desired optimal
> results for examples #16 and #56 in Chapter 1 of Timofeevs book.
>
> However this messy algorithm is not appropriate for an elegant,
> rule-based system; and it requires the host CAS provide a strong limit
> package and good algebraic simplification. So I was delighted to
> discover the following identity that makes it easy to transform
> discontinuous antiderivatives into continuous ones:
>
> If g(x) = arctan(a*tan f(x)) and
> h(x) = f(x) - arctan(cos f(x)*sin f(x) / (a/(1-a) + cos f(x)^2),
> then g(x) = h(x).
>
> Note that g(x) is discontinuous whenever f(x) mod pi equals pi/2. But
> if a>0 and f(x) is real, then the denominator a/(1-a)+cos f(x)^2 is
> never zero and so h(x) is continuous. And since the derivatives of
> g(x) and h(x) are equal, discontinuous terms in antiderivatives of the
> form g(x) can be replaced by continuous ones of the form h(x), thus
> resulting in antiderivatives continuous on the real line.
>
> There are analogous identities for inverse tangents of cotangents, and
> inverse cotangents of tangents and cotangents. Also I have
> generalized the identity to handle discontinuous antiderivatives of
> the form arctan(a+b*tan f(x)). I leave this generalization as an
>
> The next release of Rubi will take advantage of these identities to
> produce continuous trig antiderivatives, thereby making the
> computation of definite integrals easy and reliable.
>

Thanks for the explanation. I am briefly surfacing from a dive into
three-term recurrence relations, continued fractions, and fractional
linear transformations.

Your present formula goes beyond what had been announced for Rubi 3,
namely "continuitized" evaluation rules like

INT(1/(a+b*COS(x)^2), x) = 1/(SQRT(a)*SQRT(a+b))
*(x + ATAN(SIN(x)*COS(x)/(1/(1 - SQRT(a+b)/SQRT(a)) - COS(x)^2))).

Still, it doesn't seem to cover Charlwoods problem #8,

INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x)
= - 1/3*ATAN(COT(x)*COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1)),

a "continuitized" antiderivative for which is

x/3 + 1/3*ATAN(SIN(x)*COS(x)*(COS(x)^2 + 1)
/(COS(x)^2*SQRT(COS(x)^4 + COS(x)^2 + 1) + 1)).

Is this one too difficult for Rubi? Could Derive have done it?

Martin.

Date Subject Author
2/24/13 clicliclic@freenet.de
3/19/13 clicliclic@freenet.de
3/21/13 Waldek Hebisch
3/22/13 clicliclic@freenet.de
3/26/13 Waldek Hebisch
3/26/13 clicliclic@freenet.de
4/20/13 clicliclic@freenet.de
4/20/13 Nasser Abbasi
4/20/13 Rouben Rostamian
4/20/13 clicliclic@freenet.de
4/20/13 Rouben Rostamian
4/20/13 Axel Vogt
4/20/13 clicliclic@freenet.de
4/20/13 Axel Vogt
4/21/13 Axel Vogt
4/21/13 clicliclic@freenet.de
4/21/13 Waldek Hebisch
4/22/13 clicliclic@freenet.de
4/22/13 Axel Vogt
4/22/13 clicliclic@freenet.de
4/23/13 Waldek Hebisch
4/24/13 clicliclic@freenet.de
4/25/13 Waldek Hebisch
4/26/13 clicliclic@freenet.de
4/27/13 Waldek Hebisch
4/24/13 Richard Fateman
4/24/13 clicliclic@freenet.de
4/25/13 Richard Fateman
4/26/13 clicliclic@freenet.de
4/26/13 Axel Vogt
4/27/13 clicliclic@freenet.de
4/25/13 Waldek Hebisch
4/25/13 Peter Pein
4/25/13 Nasser Abbasi
4/26/13 Peter Pein
4/26/13 clicliclic@freenet.de
4/26/13 Peter Pein
4/26/13 clicliclic@freenet.de
4/26/13 Richard Fateman
4/27/13 clicliclic@freenet.de
4/27/13 Richard Fateman
6/30/13 clicliclic@freenet.de
6/30/13 Axel Vogt
7/1/13 clicliclic@freenet.de
7/1/13 Axel Vogt
7/1/13 Waldek Hebisch
7/2/13 clicliclic@freenet.de
7/2/13 clicliclic@freenet.de
7/2/13 clicliclic@freenet.de
7/2/13 Nasser Abbasi
7/2/13 Nasser Abbasi
7/4/13 clicliclic@freenet.de
7/4/13 Nasser Abbasi
7/4/13 Nasser Abbasi
7/5/13 clicliclic@freenet.de
7/5/13 Nasser Abbasi
7/9/13 clicliclic@freenet.de
7/10/13 Nasser Abbasi
7/10/13 Richard Fateman
7/10/13 Nasser Abbasi
7/10/13 clicliclic@freenet.de
8/6/13 clicliclic@freenet.de
9/15/13 Albert D. Rich
9/15/13 clicliclic@freenet.de
9/15/13 clicliclic@freenet.de
9/21/13 Albert D. Rich
9/21/13 clicliclic@freenet.de
9/22/13 daly@axiom-developer.org
9/24/13 daly@axiom-developer.org
9/30/13 daly@axiom-developer.org
9/22/13 Albert D. Rich
9/25/13 Albert D. Rich
9/25/13 Albert D. Rich
9/25/13 clicliclic@freenet.de
9/25/13 Albert D. Rich
9/26/13 Albert D. Rich
9/26/13 clicliclic@freenet.de
9/26/13 Albert D. Rich
9/29/13 clicliclic@freenet.de
10/1/13 Albert D. Rich
10/1/13 clicliclic@freenet.de
10/1/13 Albert D. Rich
10/5/13 clicliclic@freenet.de
10/5/13 Albert D. Rich
10/6/13 clicliclic@freenet.de
10/10/13 Albert D. Rich
10/10/13 Nasser Abbasi
10/11/13 clicliclic@freenet.de
11/6/13 Albert D. Rich
11/6/13 Nasser Abbasi
11/7/13 did
11/7/13 clicliclic@freenet.de
11/7/13 clicliclic@freenet.de
11/7/13 Albert D. Rich
11/12/13 clicliclic@freenet.de
11/12/13 Albert D. Rich
11/13/13 clicliclic@freenet.de
11/13/13 Albert D. Rich
11/14/13 clicliclic@freenet.de
11/14/13 Albert D. Rich
11/15/13 clicliclic@freenet.de
11/15/13 Albert D. Rich
11/16/13 clicliclic@freenet.de
11/16/13 clicliclic@freenet.de
11/21/13 Albert D. Rich
11/21/13 clicliclic@freenet.de
11/21/13 Nasser Abbasi
11/21/13 Albert D. Rich
11/21/13 Albert D. Rich
11/22/13 clicliclic@freenet.de
11/14/13 Albert D. Rich
11/15/13 clicliclic@freenet.de
11/15/13 Nasser Abbasi
11/16/13 clicliclic@freenet.de
11/16/13 Nasser Abbasi
11/7/13 did
11/7/13 clicliclic@freenet.de
4/20/13 Richard Fateman
4/21/13 clicliclic@freenet.de
4/20/13 Axel Vogt
4/20/13 clicliclic@freenet.de
4/20/13 Waldek Hebisch
4/21/13 G. A. Edgar
12/8/13 clicliclic@freenet.de
10/5/13 Albert D. Rich
10/6/13 clicliclic@freenet.de