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## [Axiom-developer] Re: Common ground

 From: Tim Daly Subject: [Axiom-developer] Re: Common ground Date: Tue, 22 Jun 2010 21:22:00 -0400 User-agent: Thunderbird 2.0.0.21 (Windows/20090302)

```Rich, Dave,

tl;dr Axiom works and shows shorter integrals in many cases.

I finished the first pass analysis of the 276 "rational" integrands.
This file was run to compare Rich's integration with Axiom's integration.
Rich is using Rule-based integration, Axiom is using algorithms.

I found 10 exact matches, that is, Axiom's output is exactly Rich's output.

I found 98 differ by a constant.

I found 136 that differ by a variable amount. However, this is due to
two things. First, there is difference in "solution basis" where sometimes
Rich uses atan where Axiom uses logs. These differences all disappear in
the derivative or under normalization. Second, the form of the result
may include negatives under logs or differing square roots. Both of these
spurious differences disappear under derivative or normalization.

All of the integrations succeeded.

Axiom is visibly shorter in 57 cases and may be shorter in more.
I have not yet written a "leaf counter" function so I don't have
a more reliable value.

There a couple things that need further work, namely
0240, 0241, 0244-0247, and 0261. I will revisit these later.

The results are at http://daly.axiom-developer.org/richrational.input.pdf
This file has a summary at the top showing several columns.

Column 1 is the integrand number.

Column 2 is Rich's integrand.
"ok" means there are no syntax errors.

Column 3 is Axiom' integrand.
"ok" means it integrated
"exact" means that the results are exactly the same.

Column 4 is the difference between Rich and Axiom.
"var" means there is a 'x' in the difference
"constant" means there is no 'x' in the difference
0 means the answers are equal

Column 5 is the derivative of the difference.
0 means that the integrals differ by at most a constant

Once the summary portion is complete the actual results are shown.
There are 5 sections per integrand,
tnnnn is the integrand
rnnnn is Rich's integral
annnn is Axiom's integral
mnnnn is the difference
dnnnn is the derivative
NOTE: sometimes Axiom has 2 results per integral. Both are listed.

I'll start the analysis of the other files shortly.

Tim

```

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