Coinduction is a major technique employed to prove behavioral properties of systems, such as behavioral equivalence. Its automation is highly desirable, despite the fact that most behavioral problems are $\Pi_2^0$-complete. Circular coinduction, which is at the core of the $\CIRC$ prover, automates coinduction by systematically deriving new goals and proving existing ones until, hopefully, all goals are proved. Motivated by practical examples, circular coinduction and $\CIRC$ have been recently extended with several features, such as special contexts, generalization and simplification. Unfortunately, none of these extensions eliminates the need for case analysis and, consequently, there are still many natural behavioral properties that $\CIRC$ cannot prove automatically. This paper presents an extension of circular coinduction with case analysis constructs and reasoning, as well as its implementation in $\CIRC$. To uniformly prove the soundness of this extension, as well as of past and future extensions of circular coinduction and $\CIRC$, this paper also proposes a general correct-extension technique based on equational interpolants.