A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $pi$ that has a density $hat{pi}$ on $mathbb{R}^d$ known up to a normalizing constant. Moreover, $-log hat{pi}$ is assumed to have a locally Lipschitz gradient and its third derivative is locally H{o}lder continuous with exponent $beta in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-log hat{pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.
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