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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: 1,138 Registered: 4/26/08
Re: The A. F. Timofeev symbolic integration test suite
Posted: Nov 13, 2013 6:56 AM

Albert Rich schrieb:
>
> On Tuesday, November 12, 2013 7:33:53 AM UTC-10, clicl...@freenet.de wrote:
>

> > I noticed these points by scanning your file visually and checking
> > against the book only where I had doubts; no systematic comparison was
> >

>
> All your proposed changes are incorporated into the revised pdf file
> at
>
> http://www.apmaths.uwo.ca/~arich/TimofeevChapter8TestResults.pdf
>
> with the exception of adding generic exponents for examples 5a and
> 5b, for which I could not find closed-form antiderivatives...
>

Oops. Integrals 5a and 6a for a generic exponent have the evaluations:

INT((a^(k*x) + a^(l*x))^n, x) =
(a^(k*x) + a^(l*x))^n/(k*n*LN(a)*(1 + a^((l - k)*x))^n)
*F21(-n, - k*n/(k - l), - k*n/(k - l) + 1, - a^((l - k)*x))

INT((a^(k*x) - a^(l*x))^n, x) =
(a^(k*x) - a^(l*x))^n/(k*n*LN(a)*(1 - a^((l - k)*x))^n)
*F21(-n, - k*n/(k - l), - k*n/(k - l) + 1, a^((l - k)*x))

The hypergeometric series here (and in Example 3) terminate for a
non-negative integer exponent n, while the series for your evaluation of
Examples 6a,b terminate (after applying Euler's transformation) for a
negative integer exponent n. The latter type of representation is
somewhat more compact; a disadvantage is that the antiderivative 6a for
positive a,m,x ends up on the branch cut of the hypergeometric function.
I therefore propose to normalize to the former type:

INT((1 + a^(m*x))^n, x) =
(1 + a^(m*x))^n/(m*n*LN(a)*(1 + a^(- m*x))^n)
*F21(-n, -n, 1 - n, - a^(- m*x))

INT((1 - a^(m*x))^n, x) =
(1 - a^(m*x))^n/(m*n*LN(a)*(1 - a^(- m*x))^n)
*F21(-n, -n, 1 - n, a^(- m*x))

I also notice now that the evaluations proposed for Examples 36a,b,c,d
involve TAN(x/2) and friends, whereas your evaluations for Examples 38,
40, 41, 43 involve SIN(x)/(1+COS(x)) and friends instead. I propose to
normalize in accordance with your policy for the entire Timofeev suite.
(In fact, I like the latter choice more; in Example 36 I simply followed
Timofeev, p. 355.)

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