  In arXiv:2204.03190, we proposed a universal method to reduce one-loop
integrals with both tensor structure and higher-power propagators. But the
method is quite redundant as it does not utilize the results of lower rank
cases when addressing certain tensor integrals. Recently, we found a remarkable
recursion relation arXiv:2203.16881,2205.03000, where a tensor integral is
reduced to lower-rank integrals and \textit{lower terms} corresponding to
integrals with one or more propagators being canceled. However, the expression
of the lower terms is unknown. In this paper, we derive this non-trivial
recursion relation for non-degenerate and degenerate cases and provides an
explicit expression for the lower terms, thus simplifying and speeding up the
reduction process.
