I will present some regularity results for vectorial minimizers of integral functionals of the type $$\int_\Omega f(x,Du(x))\,dx$$ with energy densities $f(x,\xi)=\tilde f (x,|\xi|)$ that are degenerate convex with respect to the $\xi$ variable and satisfy non standard growth conditions. Assuming that the partial map $x\mapsto f(x,\xi)$ belongs to a suitable Sobolev class, we establish the higher differentiability and the higher integrability of the gradient of the minimizers.