Date: Oct 5, 2013 9:27 PM
Author: Albert D. Rich
Subject: Re: The A. F. Timofeev symbolic integration test suite

On Saturday, October 5, 2013 1:36:36 AM UTC-10, wrote:

> You have replaced "+-" by "+" in examples #14 and #15 on p. 26. In your
> place I would prefer if Timofeev's choice of integrands (inasmuch as
> they can be inferred from the printed book at all) were not modified in
> any way, lest you open yourself to accusations of having introduced a
> selection bias which compromises the strict neutrality of the test suite
> - Timofeev couldn't have known about the systems tested, but you do.
> Hence I suggest to duplicate these integrands to "+" and "-" versions
> and to split the example numbers into 14a/b and 15a/b. In fact, integral
> tables and symbolic integrators often distinguish between expressions
> like b^2*SIN(x)^2 + a^2 and b^2*SIN(x)^2 - a^2, although the evaluations
> in these two cases are not much affected (which fact is worth noting
> too).
> This being a matter of principle, the choice of model antiderivatives is
> often just a matter of taste. In examples #16 and #56 you list the
> simpler but discontinuous ATAN evaluations. Are you retracting your
> statement of June 18 that "continuity of antiderivatives trumps
> compactness"?


> In the evaluation of example #59, -4 + 4*SIN(x)^2 may be simplified to
> -4*COS(x)^2, and the evaluation of example #70 can (accidentally?) be
> written more compactly as 1/48*SIN(2*x)^3 - 1/64*SIN(4*x) + x/16.

A revised version of the Timofeev Chapter 1 test examples and results is now available at

It incorporates all your suggested changes except for example #70. Generally I prefer antiderivatives that involve the same kernels as the integrand, even though superficially more complicated. In example #70, sin(2x) and sin(4x) do not occur in the integrand, but sin(x) and cos(x) do. However of course, systems that return antiderivatives in multiangle form should be given full credit for their answers. Like you say, it's just a matter of taste.

I did make the optimal results for examples #16 and #56 continuous as you suggested. However, all the systems tested, except Derive, return the more compact, but discontinuous antiderivative. Note that even Derive returns the discontinuous form when single-stepping through the integration. Currently, I am being lenient (and perhaps unfair to Derive) and giving full credit to all the systems for these two examples. Do you concur with this grade?