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In this paper, we present a high-order generalized extended Born approximation (Ho-GEBA) algorithm for modeling 3D electromagnetic responses of an arbitrary anisotropic body in transverse anisotropic background medium based on integral equation method. First, generalized series solutions of the integral equation are obtained by successive iterative technique, and a contraction operator is introduced for the anisotropic medium based on the iterative dissipation principle to guarantee the convergence of high-order series. Then, we derive the Ho-GEBA solutions of 3D electromagnetic responses in the anisotropic medium using the abnormal body domain decomposition method combining with the extended Born approximation. Analytical solutions of dyadic Green's functions in the transverse isotropic medium are used, which can improve the efficiency of the algorithm greatly. Numerical results show the validity of the algorithm by comparing it with the full integral equation method and the classical Born approximation.