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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: An independent integration test suite
Posted: Apr 24, 2013 12:38 PM

Waldek Hebisch schrieb:
>
> clicliclic@freenet.de wrote:

> >
> > It has occured to me that Waldek should post the FriCAS results for the
> > two cases he could not verify by differentiation. Perhaps they are
> > correct, perhaps not. I doubt that results with mixed branches should be
> > counted as successes: numerical evaluation would fail even on FriCAS, I
> > suppose.
> >

>
> (1) -> integrate((asin(x/a)*asin(x/a)^(1/2))/((-1*x^2+a^2)^(1/2)), x)
>
> +-------------------+
> x 2 | x
> 2atan(------------) |2atan(------------)
> +---------+ | +---------+
> | 2 2 | | 2 2
> \|- x + a \| \|- x + a
> (1) ------------------------------------------
> +-+
> 5\|2
> Type: Union(Expression(Integer),...)
> (2) -> integrate(acos((x/(x+1))^(1/2)), x)
>
> +-----+
> 1 | x +-----+
> (2) (x + 1)atan(----------------) + |----- \|x + 1
> +-----+ \|x + 1
> | x +-----+
> |----- \|x + 1
> \|x + 1
> Type: Union(Expression(Integer),...)
>

These are correct antiderivatives for integrands

INT(ATAN(x/SQRT(a^2 - x^2))^(3/2)/SQRT(a^2 - x^2), x) =
2/5*ATAN(x/SQRT(a^2 - x^2))^(5/2)

INT(ATAN(1/(SQRT(x/(x+1))*SQRT(x+1))), x) =
(x+1)*ATAN(1/(SQRT(x/(x+1))*SQRT(x+1))) + SQRT(x/(x+1))*SQRT(x+1)

that differ from the original ones in the omission of piecewise constant
prefactors. I guess such a rewriting of integrands in FriCAS could take
place as a preparatory step or, perhaps more likely, implicitly during
their passage through the Risch mill.

With respect to the original integrands, the first antiderivative holds
for -pi/2 < arg(a) <= pi/2 and fails for all other parameters, while the
second one fails for real x < -1 and holds everywhere else.

ch. ex's Der. Fri. Mpl MMA etc. etc.
6.10 1.20 17 ...
----------------------------------------------
1 81 78 78 80 ??
2 90 - - - -
3 14 - - - -
4 132 - - - -
5 120 - - - -
6 26 - - - -
7 11 - - - -
8 59 - - - -
--------------------------------------------
100% 96.3% 96.3% 98.8% ????%

Accordingly, the Winner so far is .............. Maple!

Martin.

PS: I noticed that the evaluation of the original first integral may
also be written without a piecewise constant in the antiderivative:

INT(ASIN(x/a)^(3/2)/SQRT(a^2 - x^2), x) =
2/5*ASIN(x/a)^(3/2)*ATAN(x/SQRT(a^2 - x^2))

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