It might help to use PiecewiseExpand here. Integrate will have an easier time dealing with explicit Piecewise objects than with various other formulations thereof.

ycc[x_] = 
 PiecewiseExpand[
  3 Max[0, (-9 + x)]^3 + 1/4 Max[0, (-9 + x)]^4 + 
   7/2 Max[0, (-3 + x)]^3 - 1/4 Max[0, (-3 + x)]^4 - 12 x^2 + c1 x + 
   c2]

(* Out[86]= Piecewise[{{c2 + c1*x - 12*x^2, x <= 3}, 
     {(1/2)*(-1323 + 2*c2 + 243*x + 2*c1*x - 33*x^2 + x^3), x > 9}}, 
   (1/4)*(-459 + 4*c2 + 486*x + 4*c1*x - 228*x^2 + 26*x^3 - x^4)] *)

Integrate[ycc[x], x, x]

(* Out[87]= Piecewise[{{(1/2)*(c2*x^2 + (c1*x^3)/3 - 2*x^4), x <= 3}, 
     {(9/2)*(-18 + c1 + c2) - (9/10)*(-36 + 5*c1 + 5*c2) + 
         (1/40)*(3321*x + 
       5*(-459 + 4*c2)*x^2 + (10/3)*(243 + 2*c1)*x^3 - 
              190*x^4 + 13*x^5 - x^6/3), 
   Inequality[3, Less, x, LessEqual, 9]}}, 
   (81/2)*(-138 + 3*c1 + c2) - (81/40)*(-549 + 60*c1 + 20*c2) + 
     (1/40)*(82053*x + 
     10*(-1323 + 2*c2)*x^2 + (10/3)*(243 + 2*c1)*x^3 - 
          55*x^4 + x^5)] *)

I was not able to figure out what it was about the solving for the constants that you did not like. In particular that NSolve formulation looked fine. If it is giving something incorrect then there needs to be a better explanation of why, and of what is the desired result.

I know how frustrating it is to loose the outcome of an hour of work. Nevertheless, I would need to know the equations which 'create' the c1 and c2.