  The anomalous Hall effect is caused by magnetic textures such as skyrmions.
We derive an analytical formula of the Hall conductivity on the surface of a
topological insulator up to third order in magnetization,
$\boldsymbol{M}(\boldsymbol{x})$, based on a perturbative approach. We identify
the magnetic textures that contribute to the Hall conductivity up to third
order in magnetization and second order in spatial differentiation. We treat
magnetization as a perturbation to calculate the Hall conductivity for each
magnetic texture based on the linear response theory. Furthermore, we estimate
the skyrmion-induced Hall conductivity and confirm that it depends on the shape
of skyrmions, such as Bloch-type or N\'eel-type skyrmions. The results of this
study can be applied not only to conventional skyrmion systems but also to more
general magnetic structures.
