In this paper, we obtain approximations for the optimal Log-Likelihood Ratio (LLR) decision rule in cooperative detection when local energy detectors are assumed. Considering conditional independence, we also show under which bandwidth and sampling frequency regimes these approximations hold best. Furthermore, we present simulations where the performance of the approximated LLR decision rule is compared with other sub- optimal decision rules given in the literature such as the optimal linear weighting. The simulations show that the density functions of the approximations exhibit negligible error in comparison with the exact ones, when conditions on bandwidth and sampling frequencies are met. The approximations presented in this paper allow to perform efficiently the joint LLR decision rule for a set of nodes without requiring Monte-Carlo simulations.