Structural and (Noetherian) cyclic induction are two instances of the Noetherian induction principle adapted to reason on first-order logic. From a theoretical point of view, every structural proof can be converted to a cyclic proof but the other way is only conjectured. From a practical point of view, i) structural induction principles are built-in or automatically issued from the analysis of recursive data structures by many theorem provers, and ii) the implementation of cyclic induction reasoning may require additional resources such as functional schemas, libraries and human interaction. In this paper, we firstly define a set of conjectures that can be proved by using cyclic induction and following a similar scenario. Next, we implement the cyclic induction reasoning in the Coq proof assistant. Finally, we show that the scenarios for proving these conjectures with structural induction differ in terms of the number of induction steps and lemmas, as well as proof scenario. We identified three conjectures from this set that are hard or impossible to be proved by structural induction.