  Coherent state functional integral for the minisuperspace model of loop
quantum cosmology is studied. By the well-established canonical theory, the
transition amplitude in the path integral representation of loop quantum
cosmology with alternative dynamics can be formulated through group averaging.
The effective action and Hamiltonian with higher-order quantum corrections are
thus obtained. It turns out that for a non-symmetric Hamiltonian constraint
operator, the Moyal (star)-product emerges naturally in the effective
Hamiltonian. For the corresponding symmetric Hamiltonian operator, the resulted
effective theory implies a possible quantum cosmological effect in large scale
limit in the alternative dynamical scenario, which coincides with the result in
canonical approach. Moreover, the first-order modified Friedmann equation still
contains the particular information of alternative dynamics and hence admits
the possible phenomenological distinction between the different proposals of
quantum dynamics.
