We study the bulk acceleration in relativistic axisymmetric magnetized outflows, by solving the momentum equation along the flow, the so-called wind equation. The solutions for the bulk Lorentz factor depend on the geometry of the field/streamlines through the "bunching function" S. We investigate the general characteristics of the S function and how its choice affects the acceleration. In our study, various fast rise and slow decay examples are selected for S, with a global maximum near the fast magnetosonic critical point, as required from the regularity condition. For each case we determine the terminal Lorentz factor γ∞ and the acceleration efficiency γ∞/μ, where μ is the total energy-to-mass flux ratio (which equals the maximum possible Lorentz factor of the outflow). With proper choices of S we can achieve efficiencies greater than 50%. Last, we examine the shape of the field/streamlines with respect to the choice of the S function. The results of this work, depending on the choices of μ, can be applied to relativistic GRB or AGN jets.