  In this work, we study the Renyi-alpha entropies S_{alpha}(\hat{rho}) = (1 -
alpha)^{-1} \ln{Tr(\hat{rho}^{alpha})} of quantum states for N bosons in the
phase-space representation. With the help of the Bopp rule, we derive the
entropies of Gaussian states in closed form for positive integers alpha =
2,3,4, ... and then, with the help of the analytic continuation, acquire the
closed form also for real values of alpha > 0. The quantity S_2(\hat{rho}),
primarily studied in the literature, will then be a special case of our
finding. Subsequently we acquire the Renyi-alpha entropies, with positive
integers alpha, in closed form also for a specific class of the non-Gaussian
states (mixed states) for N bosons, which may be regarded as a generalization
of the eigenstates |n> (pure states) of a single harmonic oscillator with n >=
1, in which the Wigner functions have negative values indeed. Due to the fact
that the dynamics of a system consisting of $N$ oscillators is Gaussian, our
result will contribute to a systematic study of the Renyi-alpha entropy
dynamics when the current form of a non-Gaussian state is initially prepared.
