The idea of hybrid homomorphic encryption (HHE) is to drastically reduce bandwidth requirements when using homomorphic encryption (HE) at the cost of more expensive computations in the encrypted domain. To this end, various dedicated schemes for symmetric encryption have already been proposed. However it is still unclear if those ideas are already practically useful, because (1) no cost-benefit analysis was done for use cases and (2) very few implementations are publicly available. We address this situation in several ways. After we formally define HHE in a broader sense than before, we build an open-source benchmarking framework involving several use cases covering three popular libraries. Using this framework, we explore properties of the respective HHE proposals. It turns out that even medium-sized use cases are infeasible, especially when involving integer arithmetic. Consequently, we propose Pasta, a cipher thoroughly optimized for integer HHE use cases. Pasta is designed to minimize the multiplicative depth, while also leveraging the structure of both state-of-the-art integer HE schemes (BFV and BGV) to minimize the homomorphic evaluation latency. Using our new benchmarking environment, we extensively evaluate Pasta in SEAL and HElib and compare its properties to 7 existing ciphers in two use cases. Our evaluations show that Pasta outperforms its competitors for HHE both in terms of homomorphic evaluation time and noise consumption, showing its efficiency for applications in real-world HE use cases. Concretely, Pasta outperforms Agrasta by a factor of up to $82$ and Masta by a factor of up to $6$ when applied to the two use cases.