The cooperative relay cognitive interference channel (RCIC) is a four-node network with two source nodes (primary source and cognitive source) and two destination nodes, in which sources try to communicate at certain rates with their corresponding destinations simultaneously through a common medium and each destination can act as a relay to assist the other one. For the partially cooperative RCIC (PC-RCIC), in which only one of the destinations (corresponding to the cognitive source) acts as a relay, we derive an achievable rate region based on using rate splitting and superposition coding at the cognitive source, and using decode-and-forward scheme at the relay. For the degraded PC-RCIC, we characterize the capacity region. We also investigate the Gaussian PC-RCIC in details. In this case, we present the achievability and converse arguments for a class of degraded Gaussian PC-RCIC and determine the capacity region of this class. Obtained results offer the cooperative relaying as an effective strategy for improving the capacity region of the cognitive interference channels (CICs).