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The average throughput of three decode-and-forward (DF) half-duplex (DFHD) and two DF full-duplex (DFFD) rateless coded relaying schemes are studied under a peak power constraint (PPC) and an average power constraint (APC). Two of the DFHD and one of the DFFD protocols are known. One new, much simpler, DFHD protocol, and one new DFFD protocol are proposed. The fairness in comparing the protocols in terms of energy cost is formulated, and the protocols are compared to each other in a power-fair regime. The two previously proposed DFHD protocols, one of which is superior to the other under the PPC, are found to have almost the same performances when compared power-fairly. The proposed protocols are built upon opportunistic communication and power optimization such that, although inferior to their predecessors in a PPC regime, they can slightly surpass them under the APC.