On Sunday, September 29, 2013 10:49:28 AM UTC-10, clicl...@freenet.de wrote:
> I have converted the 61 examples from Chapter 9 of Timofeev's book to > Derive format and made a visual comparison of your evaluations with > those in the book. Thus I discovered that two integrands (and their > antiderivatives) must be corrected: in example #45 a misprinted > (x^2-1)^(3/2) must be replaced by (x^2-1)^(5/2), and in example #46 your > ACSC(x)^2 must be replaced by ACSC(x)^4, as is actually printed. > Timofeev's evaluations correctly differentiate back to the revised > integrands. > > For reasons of quality, uniformity, conciseness, and aesthetics, I also > propose to: > - replace 2*ATANH((1 + 2*x^2)) by LN(1 + 1/x^2) in examples #4, #33, #34 > and #35 to get rid of the imaginary offset. > - replace 2*ATANH((1 - 2*x^2)) by LN(1 - 1/x^2) in example #42 for the > same reason and to get rid of bad branch cuts. > - express ACSC as ASEC in example #47 since this is the rule for all > other examples. > - change SQRT(2 - COSH(x)^2) to SQRT(1 - SINH(x)^2) in example #57 > because the second radicand looks more natural. > - move the piecewise constant x/SQRT(x^2) into ATANH and simplify to > ATANH(1/SQRT(x^2)) in examples #40, #41 (and the old #45). > - move the piecewise constant x/SQRT(x^2) into ACOTH and simplify to > ATANH(1/SQRT(x^2)) in example #43. > - remove a constant term from the evaluation of example #36 as you see > fit. > - perhaps replace SQRT(x^2)/x by x/SQRT(x^2) in examples #37, #38, #43, > #47 and #53, as the latter form is used elsewhere. In the same vain, > simplify SQRT(x^2)/x^3 to 1/(x*SQRT(x^2)) in example #38, (simplify > SQRT(x^2)/x^2 to 1/SQRT(x^2) in the old example #46), simplify > SQRT(x^2)/x^3 to 1/(x*SQRT(x^2)) and SQRT(x^2)/x^5 to 1/(x^3*SQRT(x^2)) > in example #47, and simplify SQRT(x^2)/x^2 to 1/SQRT(x^2) and > SQRT(x^2)/x^4 to 1/(x^2)^(3/2) in example #48.
Most of the changes you suggested for Chapter 9 of the Timofeev symbolic integration test suite are included in the revised test results file at
However I take issue with your assertion that Timofeev's antiderivatives for examples #45 and #46 differentiate back to the integrands. They are still off by piecewise constant factors.
I do not know what you mean by removing "a constant term from the evaluation of example #36".
Note that despite the more compact form of many of the optimal antiderivatives, the scores of the various systems on the Chapter 9 test suite did not change. This is justifiable because the post-integration simplifications required should be done automatically by the systems (but aren't...), and are not really the responsibility of the integrators.