I dug up this discussion of some integration problems in Macsyma, and a
test of cvs maxima shows they are still there for us:
http://library.wolfram.com/mathgroup/archive/1990/Nov/msg00033.html
Key points:
"I only fixed MACSYMA a few months ago to get e.g.
integrate(sqrt(x+1/x-2),x,0,1); -> 4/3, instead of -4/3 ."
Significant, nontrivial integration problem:
(c1) /* The following gives the wrong answer: */
integrate(sqrt(2-2*cos(x)),x,0,2*%pi);
(d1) 0
(c2) /* Here's the reason why: */
/* Just look at the indefinite integral and then
take limits: */
integral: integrate(sqrt(2-2*cos(x)),x);
4
(d2) - -----------------------
2
sin (x)
sqrt(------------- + 1)
2
(cos(x) + 1)
(c3) upper_limit: limit(integral,x,2*%pi,minus);
(d3) - 4
(c4) lower_limit: limit(integral,x,0,plus);
(d4) - 4
(c5) upper_limit - lower_limit;
(d5) 0
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