Read it again. I said "varying" as in "non-constant" mass, thrust, drag, density. There is no closed form solution to those trajectory equations with non-constant parameters, so unless you use numerical methods, you haven't solved them. Rocksim, OpenRocket, RasAero, even ol' RASP can model those parameters very well with numerical methods. They capture the physics better than any hand calculation possibly can, and hence are more accurate.
You said you were going for Level 3. Simple approximations to make the equations tractable don't cut it when your rocket goes high, fast, and burns a lot of propellant. My money is on RockSim.
Go read the Electronics and Software forum. There is lots of good discussion on code, models, accuracy, and interpreting flight data. Computer models are ideal for TARC students.