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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: Rubi 4.1 and the Timofeev test suite
Posted: Sep 26, 2013 1:17 PM

Albert Rich schrieb:
>
> Your perfectionist credentials remain impeccable! Chapter 9 of the
> Timofeev test suite, revised as you suggested, is now available at
>
> http://www.apmaths.uwo.ca/~arich/TimofeevChapter9TestResults.pdf
>

Wasn't as perfect as you thought! Here's the next iteration on example
#49 from Timofeev's Chapter 9:

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

I notice that Timofeev (p. 422) gives an evaluation incompatible with
the above: it doesn't differentiate back to something as simple as his
integrand, so one may conclude he made the mistake rather than the
typesetter.

As regards example #19 from Chapter 5, Timofeev gives a finite sum (p.
220) involving Sin[4*(m-p)*x] with p running from 0 to m-1 in place of
your Hypergeometric2F1[1/2-m, 1/2+m, 3/2+m, Cos[x]^2] times elementary
factors. Your hypergeometric series terminates when 2*m is an odd
integer greater than -3; and restated as Hypergeometric2F1[1+2*m, 1,
3/2+m, Cos[x]^2] times different elementary factors, it terminates when
m is a negative integer. Though not a polynomial, it is an elementary
function for any non-negative integer m as well, for m=3 one has for
instance

F21(1/2 - 3, 1/2 + 3, 3/2 + 3, z)
= 35*ASIN(SQRT(z))/(1024*z^(7/2)) + 7*SQRT(1 - z)
*(256*z^5 - 640*z^4 + 432*z^3 - 8*z^2 - 10*z - 15)/(3072*z^3)

but at present I don't know how to express this in finite terms for
arbitrary non-negative integer m. Unfortunately, non-negative integer m
are what Timofeev had in mind here. On the other hand, I think the
exponent m should not be restricted in the test suite.

As regards example #75 from Chapter 5, it seems that Timofeev meant to
write (p. 273):

INT((COS(2*x) - 3*TAN(x))*COS(x)^3
/((SIN(x)^2 - SIN(2*x))*SIN(2*x)^(5/2)), x)
= (12 - 50*TAN(x) - 135*TAN(x)^2)/(240*SQRT(2)*TAN(x)^(5/2))
+ 33/64*LN((SQRT(2) + SQRT(TAN(x)))/(SQRT(2) - SQRT(TAN(x))))

This holds where the radicands are positive; an even shorter evaluation
for the entire complex plane is:

INT((COS(2*x) - 3*TAN(x))*COS(x)^3
/((SIN(x)^2 - SIN(2*x))*SIN(2*x)^(5/2)), x)
= COS(x)/SQRT(SIN(2*x))*(1/20*COT(x)^2 - 5/24*COT(x) - 9/16)
+ 33/32*ATANH(SQRT(SIN(2*x))/(2*COS(x)))

I notice that you write SIN(2*x)^(5/2) for his SQRT(SIN(2*x)^5); I
concur that we should allow to modify integrands in this way if the
piecewise constants become simpler in consequence. However, one
shouldn't try to reinterpret radicals like SQRT(SIN(x)*COS(x)^2) in
example #80, I think.

Can you point out where you couldn't map Timofeev's evaluations to your
'optimal' ones - modulo apparent misprints and any unavoidable piecewise
constants? This would help focusing the attention of sci.math.symbolic
the book means an awful lot of work.

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