Search All of the Math Forum:

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

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

 Messages: [ Previous | Next ]
 clicliclic@freenet.de Posts: 1,099 Registered: 4/26/08
Re: Rubi 4.1 and the Timofeev test suite
Posted: Sep 25, 2013 3:06 PM

Albert Rich schrieb:
>
> Chapter 5 of Timofeev's book discusses the integration of expressions
> involving trig functions. I transcribed the 121 problems in the
> chapter and attempted to find optimal antiderivatives for them. They
> are available as a pdf file at
>
> http://www.apmaths.uwo.ca/~arich/TimofeevChapter5IntegrationProblems.pdf
>
> The problems along with all the problems in the odd-numbered chapters
> of Timofeev book are also available in machine readable form at
>
> http://www.apmaths.uwo.ca/~arich/
>
> expressed in Axiom, Maple, Mathematica and Maxima syntax.
>
> Timofeev book contains numerous typographical errors, and he does not
> seem to distinguish between (z^m)^(1/n) and z^(m/n). Also some of the
> problems require huge, non-elementary antiderivatives which were
> surely not his intent. If based on Timofeev proposed antiderivative
> you can determine his true intentions or you know of a more optimal
> antiderivative, please let me know so I can revise the test suite.
>

You have been too fast for Peter! But it looks somebody with a knack for
mathematical puzzle solving is needed now. I am still in the process of
digesting the Chapter 9 examples. Here are some more suggestions:

In example #12 replace #i*ATAN(x + #i*SQRT(1-x^2)) by ATANH(#i*x -
SQRT(1-x^2)), and similarly in #19 replace #i*ATAN(x - #i*SQRT(1-x^2))
by ATANH(#i*x + SQRT(1-x^2)). This saves one imaginary unit, and also
makes the ATANH argument a common subexpression, as also found in
examples #37 and #43.

In example #49 convert ATANH to ATAN and collapse the piecewise
constants. This gives the more natural and simpler evaluation:

INT(ASIN(SQRT((x - a)/(x + a))), x) =
- 2*a*(SQRT((x - a)/(x + a))/SQRT(2*a/(x + a)))
+ x*ASIN(SQRT((x - a)/(x + a)))
+ a*ATAN(SQRT((x - a)/(x + a))/SQRT(2*a/(x + a)))

The ATANH argument of the old evaluation is complex when the radicand
(x - a)/(x + a) is positive; such a result doesn't deserve full points.

In example #55 replace SQRT(1 - x^2) by SQRT(1 - x)*SQRT(1 + x) and
simplify, which results in:

INT(ASIN(x)/(1 - x)^(5/2), x) = - SQRT(1 + x)/(3*(1 - x))
+ 2*ASIN(x)/(3*(1 - x)^(3/2)) - SQRT(2)/6*ATANH(SQRT(1 + x)/SQRT(2))

And in example #56 move part of the piecewise prefactor into the ATANH
argument and simplify (x + 1)*SQRT(x - 1)/SQRT(x^2 - 1) to SQRT(x^2 -
1)/SQRT(x - 1) throughout the evaluation:

INT((x - 1)^(5/2)*ACSC(x), x) = 2/7*(x - 1)^(7/2)*ACSC(x)
+ 4/105*(x/SQRT(x^2))*(83 - 19*x + 3*x^2)*(SQRT(x^2 - 1)/SQRT(x - 1))
+ 4/7*(x/SQRT(x^2))*ATANH(SQRT(x^2 - 1)/SQRT(x - 1))

I saw Peter on his website giving polylogarithm antiderivatives for
examples #8 and #9 from Chapter 7, but his expressions (presumably from
Sage/Maxima) are not valid on the entire complex plane. In addition to
being shorter, the following however are:

INT(x*SIN(x)^3/COS(x), x)
= 1/4*x*COS(2*x) - 1/8*SIN(2*x) + INT(x*TAN(x), x)
= 1/4*x*COS(2*x) - 1/8*SIN(2*x)
+ #i/2*x^2 - x*LN(1 + #e^(2*#i*x)) + #i/2*DILOG(1 + #e^(2*#i*x))

INT(x*SIN(x)^3/COS(x)^3, x)
= x/(2*COS(x)^2) - 1/2*TAN(x) - INT(x*TAN(x), x)
= x/(2*COS(x)^2) - 1/2*TAN(x)
- #i/2*x^2 + x*LN(1 + #e^(2*#i*x)) - #i/2*DILOG(1 + #e^(2*#i*x))

Note that Derive's DILOG(z) corresponds to POLYLOG(2, 1-z).

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