  We show the generalization of Hsiung-Minkowski integral formula for
anisotropic capillary hypersurfaces in the half-space, which includes the
weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity
for closed hypersurfaces as special cases. As applications, we prove some
anisotropic Alexandrov-type theorems and rigidity results for anisotropic
capillary hypersurfaces. Specially, the uniqueness of the solution to the
anisotropic Orlicz-Christoffel-Minkowski problem is obtained.
