Geodesics in an Alcubierrie Warp Drive metric

The geodesics in an Alcubierrie Warp Drive metric are given by
$$dot{t} = 1 quad dot{x} = Xdot{t} quad y, z = text{constant}$$ where $X = dot{x}_s(t) f(r_s)$. Here $f(r_s)$ is a shape function given by

$$f(r) equiv frac{tanh frac{r+R}{varepsilon}-tanh frac{r-R}{varepsilon}}{2 tanh frac{R}{varepsilon}}$$ where $R$ is the radius of the warp bubble and $epsilon$ is the thickness of the warp bubble and $r_s = [(x-x_s(t))^2+y^2+z^2]^{1/2}$. Now I wish to solve the geodesic equation so naturally I have

$$x(t) = x_0 + int^t dot{x}_s(t) f(r_s) dt$$ Now, how does one evaluate this integral? Also, is $x(t)$ an increasing function of $t$ i.e. $x(t) to infty$ when $t to infty$ ? Also, what happens when we take the limit $epsilon to 0$ ?