Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?

Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(y)$, $y \in \mathbb R^n$, with the $t$ left out. With this simplifying assumption, what improvements can be seen in existing numerical methods? For example, if $n=1$, the problem turns into $t = \int \frac{dy}{f(y)}$ and we turn to a wholly different class of algorithms for integrating one-dimensional integrals. For $n>1$, the maximum possible improvement is reducing the dimension of $y$ by 1, because the time-dependent case can be simulated by appending $t$ to $y$, changing the domain of $y$ from $\mathbb R^n$ to $\mathbb R^{n+1}$.