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Topic: The Charlwood Fifty
Replies: 52   Last Post: Jun 24, 2013 10:24 PM

 Messages: [ Previous | Next ]
 Waldek Hebisch Posts: 267 Registered: 12/8/04
Re: The Charlwood Fifty, another Macsyma result
Posted: Jun 10, 2013 10:44 PM

Richard Fateman <fateman@cs.berkeley.edu> wrote:
> On 6/9/2013 7:40 PM, Albert Rich wrote:
> > On Sunday, June 9, 2013 3:37:09 PM UTC-10, Richard Fateman wrote:
> >

> >> INT(x^3*ASEC(x)/SQRT(x^4-1), x)
> >>
> >> is
> >>
> >> ((a * (sqrt(x^2 - 1) * asec(x) - 1))/2) + ((log(((a + 1)/(a -
> >> 1))))/4)
> >>
> >> with a=sqrt(x^2+1).
> >>
> >> I have not investigated how this fairs with respect to maximal
> >> continuity.

> >
> > Unfortunately, that is not a valid antiderivative since subtracting
> > its derivative from the original integrand and substituting -2 for x
> > does not equal zero.
> >
> > Albert
> >

>
> It checks out in Macsyma. In separate correspondence with Albert Rich,
> he suggested looking at how Macsyma defines the derivative of arcsecant.
> Indeed, Macsyma's definition differs by a sign for negative argument.
>
> For Macsyma's definition, the integral is correct. For Maple or
> Mathematica or Maxima, the integral needs an extra abs().
>

FYI in FriCAS:

(19) -> D(asec(x), x)

1
(19) ----------
+------+
| 2
x\|x - 1
Type: Expression(Integer)

Numerical evaluation of this at negative numbers gives wrong
value because standard branch choice for sqrt is wrong in
this case: in the formula above taking positive branch of
sqrt for x > 1 implies negative branch for x < -1. Using

1/(x^2*sqrt(1 - 1/x^2))

as formula for derivative avoids this problem. OTOH
previous formula have its own advantages, so the choice
is not so clear.

Concerning Macsyma result: for "branch correct" result
Macsyma should use a = sqrt(x^4 - 1)/sqrt(x^2-1).

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