This depends on how stable your rocket is. If your rocket is aerodynamically stable, meaning its center of pressure is behind its center of mass, the rocket will likely be turned to its velocity vector (zero angle of attack) by aerodynamics alone.
A gravity turn is optimized for the least manual maneuvering possible. Any launch trajectory besides a perfect gravity turn uses up some energy (thruster fuel or drag from fins) in forcibly changing the rocket’s velocity vector by adding to the angle of attack. Directly after launch, there is an initial small maneuver to slightly off-vertical in the direction of the turn. The acceleration due to gravity turns the rocket’s velocity vector over time, and ideally, this results in horizontal attitude at your intended orbit’s perigee. There is usually some maneuvering required to compensate for wind, turbulence, and other disturbances. The free variables involved here are the starting maneuver final attitude, thrust curves on the rocket, aero properties on the rocket, etc.
I don’t know the exact math for determining rotation rates for a particular gravity turn, but I bet it involves getting the unit direction of the earth-centered-inertial frame total acceleration of the rocket, projecting that onto the rocket’s body-yz (body-x is forward) plane, and doing a cosine for an angular rate.
If the rocket is aerodynamically unstable, with a CoP forward of the CoM, or marginally stable, with a CoP very close to the CoM, active control is required to maintain the gravity turn (usually computer guidance). This requires more energy from thrusters or fins to correct the spontaneous disturbances from the unstable aerodynamics. More unstable means more energy.
If the rocket is overstable, as described here: https://www.rocketryforum.com/threads/open-rocket-stability-number.122399/, there may be even more energy needed for course corrections due to the “weathercock” effect, the tendency to turn into the wind. Think about a dart with large fins suddenly being hit in flight with a crosswind, and how that would affect its flight path.
Excerpt from the rocket stability forum post:
I usually aim for a stability of 1.0, a stability of 1 is the Center of Gravity (CG) is ONE caliber (body tube diameter) forward of the Center of Pressure (CP). Anything less than one is considered to be marginally stable, and anything over 1.0 is considered to be over stable (iirc). Overstable rockets usually want to weather cock (turn into the wind) to varying degrees, marginally stable rockets may do everything but fly straight.