Date: Jul 19, 2013 4:47 PM
Author: clicliclic@freenet.de
Subject: Re: An independent integration test suite

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

> >
> > 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)
> >

>
> 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.
>

Derive 6.10 produces

2*LN(SQRT(b*x + 6) + SQRT(b*x + 2))/b

and

2*LN(SQRT(b*x) + SQRT(b*x + 4))/b

while Albert's model evaluations are

2*ASINH(1/2*SQRT(2 + b*x))/b

and

2*ASINH(1/2*SQRT(b*x))/b

Martin.