Date: Apr 25, 2013 12:33 PM
Author: Richard Fateman
Subject: Re: An independent integration test suite

On 4/24/2013 1:42 PM, clicliclic@freenet.de wrote:
....
>> integrate(1/(a^2+b^2*x^2),x);
>> [...]

>
> Some quick remarks on your converted suite: You have 85 entries. The
> original (like Chapter 1 of the book) has 81 items where items 14 and 15
> are vectors holding two integrals each, item 30 again holds two
> integrals, and item 48 holds three integrals. This makes a total of 86
> integrals.

yes, I found that Macsyma was unhappy with vectors that looked like
[ integral(a,x)=b=c , integral(f,x) = g = h]
and so I just put
integral(a,x)
integral(f,x)
on separate lines.
>
> Derive's #e seems to have been converted to %w (there is no %e in your
> suite).

oops. the W key is right next to the E key. I re-edited. No change in
terms of integrability. Presumably a factor of log(w) was inserted
where needed.

If we replace the string "integrate" with the string "test" in the test
file

Then define something like

test(q,v):= is (SIMPLIFY( diff(integrate(q,v),v)-q) = 0);

I got 70 confirmations, 15 were not confirmed, where SIMPLIFY was
in Maxima, a selection of transformations like ratsimp, trigsimp, and
evaluation to 0.0 at x=1.234.

This does not measure whether the form of the integral was particularly
nice, or continuous, etc. Just that it has the property of being an
antiderivative. This can matter. e.g.
integrate(x^n,x) can be expressed as (x^(n+1))/ (n+1) or
as (x^(n+1) +1) / (n+1). The latter form has the nice property that
limit as n-> -1 goes to log(x). not Infinity. (uh, plus a constant..)

I have also not tested to see if a sequence of simplification operations
can do the necessary reductions, but numerical testing with a modest
tolerance seems to confirm them all.




RJF





>
> Detailed comments tomorrow (if feasible, else later).
>
> Martin.
>