Small base stations (BSs) equipped with caching units are potential to provide improved quality of service (QoS) for multimedia service in ultra-dense cellular networks (UDCNs). In addition, the Coordinated MultiPoint (CoMP) transmission method, allowing multiple BSs to jointly serve users, is proposed to increase the throughput of cell-edge mobile terminals (MTs). Yet, the combination of content caching and CoMP in UDCNs is still not well explored for future networks. In this paper, we focus on the application of caching in CoMP-integrated UDCNs, where the cache-enabled BSs can collaboratively serve each MT using either joint transmission or single transmission. We propose a multidimensional cooperative caching (MDCC) scheme, supporting storage-dimension and transmission-dimension cooperations for the content placement. In particular, we analyze the delivery delay based on request patterns, transmission method, and the proposed cooperation strategy. Then the content placement problem is formulated as a problem of minimizing the overall expected delay. The problem is a mixed binary integer linear programming (BILP) problem, which is NP-hard. Therefore, we address the problem with approximation and substitution, and design a genetic algorithm (GA) based method to solve it. Simulation results demonstrate that the proposed MDCC scheme contributes performance gain in terms of content delivery delay in both cell-core and cell-edge areas.