This paper studies the energy efficiency and sum rate trade-off for coordinated beamforming in multi-cell multi-user multigroup multicast multiple-input single-output systems. We first consider a conventional network energy efficiency maximization (EEmax) problem by jointly optimizing the transmit beamformers and antennas selected to be used in transmission. We also account for per-antenna maximum power constraints to avoid non-linear distortion in power amplifiers and user-specific minimum rate constraints to guarantee certain service levels and fairness. To be energy-efficient, transmit antenna selection is employed. It eventually leads to a mixed-Boolean fractional program. We then propose two different approaches to solve this difficult problem. The first solution is based on a novel modeling technique that produces a tight continuous relaxation. The second approach is based on sparsity-inducing method, which does not require the introduction of any Boolean variable. We also investigate the trade-off between the energy efficiency and sum rate by proposing two different formulations. In the first formulation, we propose a new metric that is the ratio of the sum rate and the so-called weighted power. Specifically, this metric reduces to EEmax when the weight is 1, and to sum rate maximization when the weight is 0. In the other method, we treat the trade-off problem as a multi-objective optimization for which a scalarization approach is adopted. Numerical results illustrate significant achievable energy efficiency gains over the method where the antenna selection is not employed. The effect of antenna selection on the energy efficiency and sum rate trade-off is also demonstrated.