On the main page for the Rubi rule-based integrator, there is a table comparing results of Rubi against Maple and Mathematica.

On the face of it, the Maple results don't look so good. Maple is claimed as scoring about 82% correct on the author's example test suite. But only 42% of the suite gets an "optimal" result from Maple. The other 40% that Maple is claimed to get correct have results that are described as "Messy: the number of results that are correct but more than twice the size of the optimal antiderivative".

Now, I haven't scoured the tests and results to look at all the "messy" results from Maple. But a quick scan down the page of Maple results for Rational Functions shows at least one common theme: shorter forms in terms of `arctan` are prefered by the author over longer forms for the answer in terms of `ln`.

My question is this: should the length of the results really be of primary importance, given two different forms which are expressed in arctrig vs ln?

I can see that, given say a pair of results both in terms of arctrig, that expression length would be a way to compare and judge them. But if they have different types of mathematical function, then why would length be so key? And why isn't continuity mentioned?

Here's one example (amongst many that are similarly judged) from that page of Maple results:

Valid but unnecessarily complicated antiderivative:
Integrand = 1/x/(2+3*x)
Optimal integral = -arctanh(1+3*x)
Maple integral = 1/2*ln(x)-1/2*ln(2+3*x)

Do these two plots show a better criterion for judging that pair of candidate results?

plots:-complexplot3d( -arctanh(1+3*x), x=-1-I..1+I, axes=box );
plots:-complexplot3d( 1/2*ln(x)-1/2*ln(2+3*x), x=-1-I..1+I, axes=box );

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