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The integration between the satellite network and the terrestrial network will play a key role in the upcoming sixth-generation (6G) of mobile cellular networks thanks to the wide coverage and bandwidth offered by satellite networks. To leverage this integration, we propose a proactive traffic offloading scheme in an integrated multi-satellite terrestrial network (IMSTN) that considers the future networks' heterogeneity and predicts their variability. Our proposed offloading scheme hinges on traffic prediction to answer the stringent requirements of data-rate, latency and reliability imposed by heterogeneous and coexisting services and traffic namely enhanced mobile broadband (eMBB), massive machine-type communications (mMTC) and ultra-reliable low latency communication (URLLC). However, the fulfilment of these requirements during offloading in dynamic IMSTN comes at the expense of significant energy consumption and introduces inherently supplementary latency. Therefore, our offloading scheme aims to balance the fundamental trade-offs first between energy consumption and the achievable data-rate and second between energy consumption and latency while meeting the respective needs of the present traffic. Our findings prove the importance of the cooperation between the multi-satellite network and the terrestrial network conditioned by traffic prediction to enhance the performance of IMTSN in terms of latency and energy consumption.