  The studies of Ram\'irez, Hino-Ram\'irez, and Ariyoshi-Hino showed that an
integrated version of Varadhan's asymptotics holds for Markovian semigroups
associated with arbitrary strong local symmetric Dirichlet forms. In this
paper, we consider non-symmetric bilinear forms that are the sum of strong
local symmetric Dirichlet forms and lower-order perturbed terms. We give
sufficient conditions for the associated semigroups to have asymptotics of the
same type.
