integrand := (1 - (t - x1) * exp(-(t - x1)) - exp(-(t - x1))) * (1 - exp(-(t - x6))) *
             exp(-x1) * exp(-x2) * exp(-x3) * exp(-x4) * exp(-x5) * exp(-x6);
F1 := int(int(int(int(int(int(integrand, x6 = 0 ..  t - x4 - x1 - x3), x5 = 0 .. t - x1 - x3),
      x4 = 0 .. t - x1 - x3), x3 - 0 .. t - x1),  x2 = 0 .. x1), x1 = 0 .. t);
F2 := int(int(int(int(int(int(integrand, x6 = 0 .. t - x4 - x2), x5 = 0 .. t - x2),
      x4 = 0 .. t - x2), x3 = 0 .. x2 - x1), x2 = x1 .. t), x1 = 0 .. t);
F3 := int(int(int(int(int(int(integrand, x6 = 0 .. t - x4 - x1 - x3), x5 = 0 .. t - x1 - x3),
      x4 = 0 .. t - x1 - x3), x3 = x2 - x1 .. t - x1), x2 = x1 .. t), x1 = 0 .. t);
F6T := simplify(F1 + F2 + F3);
mean := int(1 - F6T, t = 0 .. infinity);