A Christmas present for your favorite CAS

[Henri Cohen]

Looking in my old files, I found the following INDEFINITE integral
(maple notation)

int(x/sqrt(x^4+10*x^2-96*x-71),x);

Of course this is an elliptic integral. However, it happens that this
special integral can be computed explicitly. Questions:

1) Can any CAS compute this (not leaving the result with elliptic functions
of course)? You are allowed to load any standard library you like.

2) Can YOU compute this?

3) Find other non-trivial examples.

Note: the experts in the field will know that there is a beautiful and rich
theory behind this kind of computable elliptic integrals. In particular,
relations with points of finite order on elliptic curves, and periodic
continued fraction expansions. This can be considered as a (admittedly
obscure) hint for non-experts.

Henri Cohen

[Manuel Bronstein]

In article <2f6ogd$s...@graffiti.cribx1.u-bordeaux.fr>,

Henri Cohen <co...@ceremab.u-bordeaux.fr> wrote:
>
>Looking in my old files, I found the following INDEFINITE integral
>(maple notation)
>
>int(x/sqrt(x^4+10*x^2-96*x-71),x);
>
>Of course this is an elliptic integral. However, it happens that this
>special integral can be computed explicitly. Questions:
>
>1) Can any CAS compute this (not leaving the result with elliptic functions
>of course)? You are allowed to load any standard library you like.
>

No need to load any outside package in Axiom:
(1) ->integrate(x/sqrt(x**4+10*x**2-96*x-71),x)

(1)
-
log
+--------------------+
6 4 3 2 | 4 2 8
(x + 15x - 80x + 27x - 528x + 781)\|x + 10x - 96x - 71 - x
+
6 5 4 3 2
- 20x + 128x - 54x + 1408x - 3124x - 10001
/
8
Type: Union(Expression Integer,...)

-- Verifying is easy
(2) ->D(%,x)

x
(2) -----------------------
+--------------------+
| 4 2
\|x + 10x - 96x - 71
Type: Expression Integer

>2) Can YOU compute this?
>

Do you mean by hand?

>3) Find other non-trivial examples.
>

- take any elliptic curve (y := sqrt(p(x)) where p(x) is squarefree and has
degree 3 or 4)

- take a rational function of f(x,y) of x and y which is not a pure power of
another such rational function, and any integer n which is not -1 or +1,
then df/dx / (n f) is a non-trivial example.
The larger the absolute value of n, the longer it will take to compute back
the integral (usually).

-----------------------------------------------------------------------------
____________
/ / / / Manuel Bronstein
/--- / /___/ bron...@inf.ethz.ch
/ / / / Informatik, ETH Zuerich, Switzerland
---- / / / Tel: [41] (1) 632-7474
Fax: [41] (1) 262-3973
-----------------------------------------------------------------------------