We present minimal integrity bases for all common anisotropies in hyperelasticity via the structural tensor concept, which can be used to formulate any algebraic invariant function in the elements of the respective bases. Hence, the provided minimal integrity bases are of great interest for formulating a concise but general anisotropic material model. Our work covers results for the 11 types of anisotropy that arise from the classical 7 crystal systems, as well as findings for 4 additional non-crystal anisotropies derived from the cylindrical, spherical, and icosahedral symmetry systems. By using well-known results from literature about structural tensors, isotropic invariants, and isotropic extension, functional bases are directly determined. A simple analytical-numerical approach is employed to identify polynomial relations between the invariants of these functional bases, thereby enabling the construction of functional bases of reduced cardinality. After that, we show that the determined reduced functional bases are also minimal integrity bases by identifying polynomial relations to known integrity bases from literature. Furthermore, fundamental concepts from invariant theory, including the Hironaka decomposition of invariant rings and the closely related Hilbert series, are employed to further validate the results. Alongside the presented findings, this article also aims to provide an introductory overview of the complex field of modeling anisotropic materials, especially for researchers with an engineering background.