On integral closedness of formal power series ring over an integrally closed domain satisfying Krull intersection principle

Question

Let $R$ be a normal domain (i.e. an integral domain integrally closed in its fraction field) such that for every non-unit $t\in R$, $\cap_{n\ge 1} (t^n)=(0)$ ; then is it true that $R[[X]]$ is normal (i.e. integrally closed in its fraction field) ?

If this is not true in general, what if we strengthen the hypothesis to say $\cap_{n\ge 1} J^n=(0)$ for every proper ideal $J$ of $R$ ?