In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb F$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the structure $Q + R$ where $Q$ is a finite quadratic variation process and $R$ is {\it{strongly predictable}} in some technical sense: that condition implies in particular that $R$ is \textit{weak Dirichlet}, and it is fulfilled, for instance, when $R$ is independent of $M$. The method is based on a transformation which reduces the {{\it diffusion}} coefficient multiplying $\xi$ to 1. We use generalized Itô and Itô-Wentzell type formulae. A similar method allows to discuss existence and uniqueness theorem when $\xi$ is a Hölder continuous process and $\sigma$ is only Hölder in space. Using an Itô formula for {\it{reversible}} semimartingales we also show existence of a solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.