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As usual, we observe an unknown coupling function, i.e. $F(\varphi)$, with afunction of torsion and also curvature, i.e. $f(T)$ and $f(R)$, generallydepending on a scalar field. In $f(R)$ case, it comes from quantum correlationsand other reasons. Now, what if beside this term in $f(T)$-gravity context, weenhance the action through another term which depends on both scalar field andits derivatives? In this paper, we have added such an unprecedented term in thegeneric common action of $f(T)$-gravity such that in this new term, an unknownfunction of torsion has coupled with an unknown function of both scalar fieldand its derivatives. We have explained why we can append such term, in theintroduction in details. By the use of Noether symmetry approach, we haveconsidered its behavior and effect. We have shown that it does not produce ananomaly, but rather it works successfully and numerical analysis of the exactsolutions of field equations coincide with all most important observationaldata particularly late-time-accelerated expansion. So, this new term may beadded to the gravitational actions of $f(T)$-gravity.

Studying the Intervention of an Unusual Term in $f(T)$ Gravity via Noether Symmetry