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GW 170817 event gave rise to a new constraint on cosmology wherein the speed of gravitational waves is approaching the speed of light. As a result, several modified gravity theories were no more viable, including the Einstein-Gauss-Bonnet theory. However, some work has shown that this theory can still be compatible when the constraint is applied, c2T ' 1. In this work, we add non-minimal coupling and non-minimal derivative coupling to the Einstein-Gauss-Bonnet inflation theory. The slow-roll approximations are applied to the background field equations. In the perturbative view, the action is expanded to the second order within the framework of Horndeski’s theory. The constraint equation is extracted from the tensor perturbation dynamics. The Gauss-Bonnet coupling function model includes linear, power and exponential coupling functions. Free parameter values in the model are evaluated to attain compatibility with the observations. Each model provides different characteristic regarding the parameters that regulate proper alterations in the value of the ns and r indices. The linear model is controlled by the non-minimal coupling constant, the potential integration constant and the Gauss-Bonnet coupling constant. The power law model is only controlled by the power value of the Gauss-Bonnet coupling function. The exponential model is controlled by the integration constant of the potential equation and the power value of the Gauss-Bonnet coupling function.Finally, former approximations are investigated to test the validity of the model. It turned out that power law and exponential models are invalid so that several equations are updated until these models are compatible