Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: The A. F. Timofeev symbolic integration test suite
Posted: Nov 12, 2013 12:33 PM

Albert Rich schrieb:
>
> Chapter 8 of Timofeev's book discusses the integration of expressions
> involving exponentials, logarithms and hyperbolic functions. I
> transcribed the 109 example problems in the chapter. The integrands
> and candidate optimal antiderivatives are available as a pdf file at
>
> http://www.apmaths.uwo.ca/~arich/TimofeevChapter8TestResults.pdf
>
> The problems from Chapter 8 and the other chapters of Timofeev's book
> are also available in machine readable form at
>
> http://www.apmaths.uwo.ca/~arich/
>
> expressed in Axiom, Maple, Mathematica and Maxima syntax. If you
> find and would like to contribute significantly better
> antiderivatives than the ones shown in the pdf file, please post them
> on sci.math.symbolic.
>

I propose to add evaluations for generic exponents n to Examples 3, 5a,
5b, 6a, 6b (p. 346), as was done for Example 97 (p. 377); such as:

INT((#e^x - #e^(-x))^n, x) = - 1/n*(#e^x - #e^(-x))^n
/(1 - #e^(2*x))^n*F21(-n, - n/2, 1 - n/2, #e^(2*x))

The evaluation of Example 8 becomes somewhat more compact when written
the Timofeev way:

INT(#e^x/(a*#e^(3*x) + b), x) =
LN(#e^x*a^(1/3) + b^(1/3))/(2*a^(1/3)*b^(2/3))
- LN(a*#e^(3*x) + b)/(6*a^(1/3)*b^(2/3))
+ SQRT(3)/(3*a^(1/3)*b^(2/3))
*ATAN((2*#e^x*a^(1/3) - b^(1/3))/(SQRT(3)*b^(1/3)))

Example 12a,b should better be repaired as follows:

INT((a + b*#e^(n*x))^(r/s)*#e^(n*x), x)
= s/(b*n*(s + r))*(a + b*#e^(n*x))^((s + r)/s)

The duplication +-r can be omitted since r is allowed to be negative.
Example 14 involves eighth (4 occurences) or sth (1 occurence) roots; in
view of Example 12, I suggest to use s (it looks like the typesetter
needed better eyeglasses). Again the duplication +-r can be omitted.

The hypergeometric evaluation of Example 33 (p. 355) can be simplified
to:

INT(#e^(m*x)*TAN(x)^2, x) = - #e^(m*x)/m + 4/(2*#i + m)
*#e^((2*#i + m)*x)*F21(2, 1 - #i*m/2, 2 - #i*m/2, - #e^(2*#i*x))

Examples 36a,b,c,d have the hypergeometric evaluations:

INT(#e^x/(1 + COS(x)), x) = #e^x*(TAN(x/2) + #i
- 2*#i*F21(1, -#i, 1 - #i, - #e^(#i*x)))

INT(#e^x/(1 - COS(x)), x) = #e^x*(- COT(x/2) + #i
- 2*#i*F21(1, -#i, 1 - #i, #e^(#i*x)))

INT(#e^x/(1 + SIN(x)), x) = #e^x*(TAN(x/2 - pi/4) + #i
- 2*#i*F21(1, -#i, 1 - #i, #i*#e^(#i*x)))

INT(#e^x/(1 - SIN(x)), x) = #e^x*(- COT(x/2 - pi/4) + #i
- 2*#i*F21(1, -#i, 1 - #i, - #i*#e^(#i*x)))

I suggest to replace #i by -#i in the evaluation of Example 43 for
conformity with Examples 38, 40, and 41.

Integrand 61 (p. 365) was meant to read 1/(a +- b*COSH(x)), but the
duplication is pointless once b is allowed to be negative. Isn't it
better here to write SQRT(a^2 - b^2) for SQRT(a - b)*SQRT(a + b) inside
and outside ATANH since a^2 - b^2 is positive when both a - b and a + b
are not? The duplication of Example 63 is pointless for your result, but
not for Timofeev's evaluation - provided he got it right, which I didn't
check.

Timofeev's evaluation of Example 93b (p. 377) is funny.

The integrand 96a must have been meant to involve SQRT(a^2 +- LN(x)^2);
the cases SQRT(LN(x)^2 - a^2) of Example 95a and SQRT(a^2 + LN(x)^2) of
Example 96a are omitted in your suite. I propose to split these
integrands into a1 and a2 (or similar).

I noticed these points by scanning your file visually and checking
against the book only where I had doubts; no systematic comparison was
made - but should eventually be made.

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