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Topic: An independent integration test suite
Replies: 42   Last Post: Jul 25, 2013 6:09 PM

 Messages: [ Previous | Next ]
 Waldek Hebisch Posts: 267 Registered: 12/8/04
Re: An independent integration test suite
Posted: Jul 19, 2013 3:37 PM

clicliclic@freenet.de wrote:
>
> daly@axiom-developer.org schrieb:

> >
> > Albert,
> >
> > I'm getting different answers than you for some of the problems.
> > In Axiom, if I do
> >
> > t0:=1/((1-x)^(9/2)*(1+x)^(5/2))
> > t1:=integrate(t0,x)
> > t2:=D(t1,x)
> > t3:=t0-t2
> >
> > I get 0 as a result. However, when I take the derivative of your
> > result and difference it from your initial equation I get a
> > non-constant result dependent on x.
> >
> > Perhaps you could check your answer in other systems and let me
> > know if you agree.
> >
> > I will be posting the first set of results shortly.
> >

>
> This integrand appears on page 1102 of rich1b.input.pdf. Its
> antiderivative
>
> - (8*x^5 - 16*x^4 - 4*x^3 + 24*x^2 - 9*x - 6)
> /(21*(1 - x)^(7/2)*(x + 1)^(3/2))
>
> is simply an algebraic function, so should be behave the same on all
> systems. Indeed your expression d000 at the bottom of page 1103
> simplifies to zero for arbitrary complex x.

The result:

r0000:=1/7/((1-x)^(7/2)*(1+x)^(3/2))+1/7/((1-x)^(5/2)*(1+x)^(3/2))+_
4/21*x/(1-x^2)^(3/2)+8/21*x/sqrt(1-x^2)

is clearly suboptimal: the integrand contains sqrt(1-x) and sqrt(x+1)
so sqrt(1-x^2) should not appear in "optimal" result.

> You are having more serious problems with the integrands on pages 994,
> 996, 998, 1000, 1004, 1008, 1017, 1019, 1020, 1021, 1022, 1107, 1109,
> and maybe elsewhere. Can FriCAS handle these (correctly) already?
>

They look like variations of single example. For the first two
FriCAS gets:

(5) -> integrate(1/(sqrt(2+b*x)*sqrt(6+b*x)), x)

+-------+ +-------+
log(\|b x + 2 \|b x + 6 - b x - 4)
(5) - -----------------------------------
b
Type: Union(Expression(Integer),...)
(6) -> D(%, x)

1
(6) --------------------
+-------+ +-------+
\|b x + 2 \|b x + 6
Type: Expression(Integer)
(7) -> integrate(1/(sqrt(b*x)*sqrt(4+b*x)), x)

+---+ +-------+
log(\|b x \|b x + 4 - b x - 2)
(7) - -------------------------------
b
Type: Union(Expression(Integer),...)
(8) -> D(%, x)

1
(8) ----------------
+---+ +-------+
\|b x \|b x + 4
Type: Expression(Integer)

In general, FriCAS is supposed to handle all integrands of similar
form. Sometimes for similar problems FriCAS gets very messy
result (Charlwood suite has a few such examples). I am
working on a cure.

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