  In this work we study the cosmological perturbations in
Bahamonde-Dialektopoulos-Levi Said (BDLS) theory, i.e. the teleparallel analog
of Horndeski gravity. In order to understand the evolution of structure in a
cosmological model, it is necessary to study its cosmology not only in the
background but also perturbatively. Both Horndeski and its teleparallel analog
have been analyzed a lot in the literature, but in order to study them
quantitatively, we need to know their cosmological perturbations. That is why,
we study here the scalar-vector-tensor decomposition of the theory and we also
express the so-called alpha parameters in terms of the arbitrary functions of
the theory, that designate the deviation from the {\Lambda}CDM model. We have
explored tensor, vector and scalar perturbation of the action up to second
order, which drastically opens up new possibilities on searches in the
parameter space of scalar-tensor theories in the context of observations.
