Integral formula for codimension-one foliated $(\alpha,\beta)$-spaces

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. Recently, we associated a new Riemannian metric to a codimension-one foliated Finsler space and proved integral formulae for general and for Randers spaces. In the paper, we study this metric for a wider class of codimension-one foliated $(\alpha,\beta)$-spaces and embody it in a set of metrics that can be viewed as a perturbed metric associated with $\alpha$. For such metrics we calculate the Weingarten operator of the leaves and deduce the Reeb type integral formula, which can be used for $(\alpha,\beta)$-spaces, e.g. Randers and Kropina spaces.