I used Wolfram alpha for Mathematica?s answer.
If there was a diracdelta() function in Maxima would integrate(unit_step(x)*diracdelta(x), x, minf, inf) equal 1 or 1/2? I think 1/2. Then there is the case of integrate(between(x,0,0,?closed)*diracdelta(x),x,minf,inf). Should that be zero?
FWIW. Maple help says that integrate(f(x) * diradelta(x), x, minf, inf) is only defined for functions f(x) that are smooth (infinitely differentiable) near 0.
Mathematica says 1 for the integrate[DiracDelta(x) * unitstep[x],x,-1,1] case and integrate[DiracDelta(x) * Heaviside[x],x,-1,1] is undefined.
Rich