Could be better, but not wrong:
(%i1) integrate(abs(x)/(1 + abs(x)), x, -2 , 2);
(%o1) integrate(abs(x)/(abs(x)+1),x,-2,2)
I think Maxima should ignore assumptions on dummy variables:
(%i2) assume(x > 0, x < %pi);
(%o2) [x>0,x<%pi]
(%i3) integrate(abs(x)/(1 + abs(x)), x, -2 , 2);
Is zeroa positive, negative, or zero?pos;
I suppose abs(x) / (1 + abs(x)) --> x / (1 + x); after that, integrate from -2 to 2. Ouch.