Computing product of matrix exponential with vector using Runge Kutta

Question

I have a matrix $A$ and a vector $b$. I need to numerically calculate $e^{t A}b$.

That can be seen as the solution of the initial value problem

$$ \frac{dy}{dt} = A y; \;\;\; y(0)=b $$

Therefore I can use Runge Kutta methods to calculate $e^{t A}b$.

Is it inefficient? What are more efficient ways to compute that? I suppose Pade approximant should be more efficient. But how much more?

Thanks!