Studying the intervention of an unusual term in f(T) gravity via the Noether symmetry approach

As has been done before, we study an unknown coupling function, i.e. $$F(\varphi )$$ , together with a function of torsion and also curvature, i.e. f(T) and f(R), generally depending upon a scalar field. In the f(R) case, it comes from quantum correlations and other sources. Now, what if beside this term in f(T) gravity context, we enhance the action through another term which depends upon both scalar field and its derivatives? In this paper, we have added such an unprecedented term in the generic common action of f(T) gravity such that in this new term, an unknown function of torsion has coupled with an unknown function of both scalar field and its derivatives. We explain in detail why we can append such a term. By the Noether symmetry approach, we consider its behavior and effect. We show that it does not produce an anomaly, but rather it works successfully, and numerical analysis of the exact solutions of field equations coincides with all most important observational data, particularly late-time-accelerated expansion. So, this new term may be added to the gravitational actions of f(T) gravity.