With commercial Macsyma, one could handle some integrands that
involve max by trading abs for max
(c1) xmax(a,b) := (a+b + abs(a-b)) / 2$
(c2) integrate(xmax(a-x,0),x,0,b);
Is a positive, negative, or zero?
pos;
Is b-a positive, negative, or zero?
pos;
a^2/2
Unfortunately, Maxima doesn't know the antiderivative of the absolute
value
function . : --(
Barton