I'm not sure what the relevance is of DERIVE's performance, since the program is neither supported nor distributed at this time, but the syntax looked almost identical to Macsyma (commercial, not supported or distributed) so I tried to run it through Macsyma 2.4.0
textual changes ... LN -> LOG, PI -> %pi, #e-.%e, INT( -> integrate(, removal of [ ].. downcasing of all trig functions and sqrt.
1. I found expressions in the test suite of the form
integrate(A,x) = B = C
which do not parse properly since B=C makes sense in Macsyma if B and C are algebraic. It does not make sense in Macsyma to ask (A=B) = C since A=B is logical not algebraic.
That is almost beside the point though. What is the test suite supposed to be used for? Testing the simplification of B=C? Is either answer, B or C supposed to be OK for the integration? Is one supposed to verify the answer by differentiating B?
So I removed all the "answers" of several forms. I am optimistically assuming that all results from integration of the forms in this chapter will pass some version of differentiation + simplification.
Running through the examples, Macsyma paused in a number of places to ask (for example) is a*b pos or neg; is a nonzero; is x >0, is n+1 nonzero..
So I added assumptions such as assume(a>0, b>0, x>0 n>0) and ran the test again.
The test went through without any burps that I noticed except for
integrate(asin(x/a)^(3/2)/sqrt(a^2-x^2),x) which was not integrated. This appears to be the same as the first clumsily restated expression in the original posting.