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

 Messages: [ Previous | Next ]
 Waldek Hebisch Posts: 267 Registered: 12/8/04
Re: An independent integration test suite
Posted: Apr 27, 2013 1:57 PM

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

> >
> > clicliclic@freenet.de wrote:

> > >
> > > Waldek Hebisch schrieb:

> > > >
> > > > (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.

> >
> > Note that as real functions asin(x/a) and atan(x/sqrt(a^2 - x^2)) are
> > defined on the same domain (that is for a > 0 interval [-a, a]) and
> > take the same values.
> >
> > The same for acos((x/(x+1))^(1/2)) and atan((x/(x+1))^(1/2)*(x+1)^(1/2)).
> >
> > FriCAS assumes that functions to integrate are real and ATM uniformly
> > performs such changes.
> >
> > You of course may have your own opinion, but IME for complex
> > integration paths one usualy wants to choose version which
> > is analytic along the path (if possible). This usually
> > requires separate resoning.

>
> Even when the integration variable and parameter 'a' are restricted to
> the real axis, neither of these two antiderivatives can be verified by
> differentiation, on FriCAS or elsewhere. But it is up to you to decide
> if they are integrated 'correctly' within the framework of FriCAS, of
> whose 'peculiarities' users should be made aware. They will happily be
> accepted as correct if you count them as such. Nobody can deny that
> FriCAS integrates them in some sense.
>

A little correction: equivalence of asin(x/a) and atan(x/sqrt(a^2 - x^2)
requires a > 0.

Concerning correctness: I consider such results as suboptimal,
but correct. Suboptimal, because normally FriCAS tries to
express result in terms of functions in the user input.
In this case FriCAS used different functions without real
need.

BTW: Core integrator normally only deals with algebraics,
exponential and logarithms, sometimes with tangent (when
the integrand is a rational function of integration variable
and a single tangent). What users see is the result of
a postprocessor. I will probably modify postprocessor
to restore asin-s and acos-es.

--
Waldek Hebisch
hebisch@math.uni.wroc.pl

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