Given a fixed network of routers, a set of multicast sources and their corresponding receivers, we investigate the problem of constructing multicast sessions that maximize the multicast throughput of all sessions under fairness constraints. It is known that for problems with only one source node, heuristic algorithms based on packing maximum-rate Steiner trees may achieve throughput close to network capacity for some networks of interest. In almost all practical applications, however, multiple multicast sessions must be concurrently supported by the same network. We find that greedy strategies such as maximum-rate Steiner tree packing fail to perform well in dense problems, where the number of sources are large. We then propose a heuristic round-robin algorithm, called Cooperative Shortest Path Tree Packing Algorithm (CSPT), that performs uniformly well in the whole spectrum of problems from sparse to dense. Simulations on random networks show up to 5 times increase in throughput when compared to conventional methods in which there is only one tree per multicast session, and on average achieving 92% of the network capacity, when network coding is allowed. Finally, we show how CSPT can be implemented, with relative ease, on top of the current standard IP protocols.