Suppose I know a function $f(t)$ and all its derivatives in $t$ in closed form. Given $a(0)$ and some $t_0>0$, I'm looking for an explicit integrator that can estimate $a(t_0)$, where $a(\cdot)$ satisfies the ODE: $$\frac{da(t)}{dt}=e^{-a(t)}f(t).$$ I'm willing to compute (in closed-form) any derivatives $f^{(k)}(t)$ if that helps.
Of course I can apply the usual explicit methods (forward Euler, RK3), but I'm hoping to find an integrator that's reversible in time. And, ideally, one that exploits special structure in the ODE above, e.g. by somehow integrating the exponential, to achieve higher accuracy on some model problems.
Any suggestions?