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In this paper, we propose novel Hybrid Automatic Repeat re-Quest (HARQ) strategies used in conjunction with hybrid relaying schemes, named as H2-ARQ-Relaying. The strategies allow the relay to dynamically switch between amplify-and-forward/compress-and-forward and decode-and-forward schemes according to its decoding status. The performance analysis is conducted from both the spectrum and energy efficiency perspectives. The spectrum efficiency of the proposed strategies, in terms of the maximum throughput, is significantly improved compared with their non-hybrid counterparts under the same constraints. The consumed energy per bit is optimized by manipulating the node activation time, the transmission energy and the power allocation between the source and the relay. The circuitry energy consumption of all involved nodes is taken into consideration. Numerical results shed light on how and when the energy efficiency can be improved in cooperative HARQ. For instance, cooperative HARQ is shown to be energy efficient in long distance transmission only. Furthermore, we consider the fact that the compress-and-forward scheme requires instantaneous signal to noise ratios of all three constituent links. However, this requirement can be impractical in some cases. In this regard, we introduce an improved strategy where only partial and affordable channel state information feedback is needed.