  We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes
exponentially whenever the stable and unstable bundles are not jointly
integrable. This allows us to show that if a flow is sufficiently close to a
volume-preserving Anosov flow and $\operatorname{dim} \mathbb{E}_s = 1$,
$\operatorname{dim} \mathbb{E}_u \geq 2$ then the flow mixes exponentially
whenever the stable and unstable bundles are not jointly integrable.This
implies the existence of non-empty open sets of exponentially mixing Anosov
flows. As part of the proof of this result we show that $\mathcal{C}^{1+}$
uniformly-expanding suspension semiflows (in any dimension) mix exponentially
when the return time in not cohomologous to a piecewise constant.
