We consider the elliptic partial differential equation in the divergence form \[ -{\rm div}(\nabla G(\nabla u(x)))+ F_u(x,u(x))=0, \] where $G$ is a convex, anisotropic function satisfying certain growth and ellipticity conditions. We prove that weak solutions in $W^{1,G}$ are in fact of class $W^{2,2}_{\rm loc}\cap W^{1,\infty}_{\rm loc}$.