A key component of Fuzzy Rule-Based Classification Systems (FRBCSs) is the Fuzzy Reasoning Method (FRM), since it infers the class predicted for new examples. A crucial stage in any FRM is the way in which the information given by the fired rules during the inference process is aggregated. A widely used FRM is the winning rule, which applies the maximum to accomplish this aggregation. The maximum is an averaging operator, which means that its result is within the range delimited by the minimum and the maximum of the aggregated values. Over the last years, new averaging operators based on generalizations of the Choquet integral were also proposed to perform this aggregation process. However, the most accurate FRBCSs use the FRM known as additive combination, that considers the normalized sum as the aggregation operator, which is non-averaging. For this reason, this paper is aimed at introducing a new non averaging operator named $C_{F1 F2}$ -integral, which is a generalization of the Choquet-like Copulabased integral (CC-integral). $C_{F1 F2}$ -integrals present the desired properties of an aggregation-like operator, since they satisfy appropriate boundary conditions and have some kind of increasingness property. We show that $C_{F1 F2}$ -integrals, when used to cope with classification problems, enhance the results of the previous averaging generalizations of the Choquet integral and they provide competitive results (even better) when compared with state-of-the-art FRBCSs.