# Hyperbolic motion with non-zero initial velocity

The equations of hyperbolic motion in a single spatial direction with initial velocity of zero are pretty easy to find given a bit of googling, such as in this answer. And while I'm too rusty at integration to derive them myself, I think I understand the general concepts behind them. However I can't seem to find the same equations for non-zero initial velocity.

Specifically I'm looking for the following equations, but for non-zero initial velocity: \begin{align} v(t) = \frac{ctA}{\sqrt{c^2 + t^2A^2}} \end{align} \begin{align} x(t) = x_0+\frac{c}{A}\left(\sqrt{c^2+t^2A^2}-c\right) \end{align} ($$A$$ is proper acceleration, $$c$$ is the speed of light, $$t$$ is coordinate time. See the linked answer for derivation of these equations.)

Using $$u_0, \gamma_0, X_0, T_0, \tau_0$$ as the initial velocity, Lorentz factor, space and time coordinates:
\begin{align}u(T) & =\frac{u_{0}\gamma_{0}+\alpha T}{\sqrt{1+\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)^{2}}}\quad\\ & =c\tanh\left\{ \sinh^{-1}\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)\right\} \\ X(T) & =X_{0}+\frac{c^{2}}{\alpha}\left(\sqrt{1+\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)^{2}}-\gamma_{0}\right)\\ & =X_{0}+\frac{c^{2}}{\alpha}\left\{ \cosh\left[\sinh^{-1}\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)\right]-\gamma_{0}\right\} \\ c\tau(T) & =c\tau_{0}+\frac{c^{2}}{\alpha}\ln\left(\frac{\sqrt{c^{2}+\left(u_{0}\gamma_{0}+\alpha T\right){}^{2}}+u_{0}\gamma_{0}+\alpha T}{\left(c+u_{0}\right)\gamma_{0}}\right)\\ & =c\tau_{0}+\frac{c^{2}}{\alpha}\left\{ \sinh^{-1}\left(\frac{u_{0}\gamma_{0}+\alpha T}{c}\right)-\tanh^{-1}\left(\frac{u_{0}}{c}\right)\right\} \end{align}
\begin{align}u(\tau) & =c\tanh\left\{ \tanh^{-1}\left(\frac{u_{0}}{c}\right)+\frac{\alpha\tau}{c}\right\} \\ \\ X(\tau) & =X_{0}+\frac{c^{2}}{\alpha}\left\{ \cosh\left[\tanh^{-1}\left(\frac{u_{0}}{c}\right)+\frac{\alpha\tau}{c}\right]-\gamma_{0}\right\} \\ \\ cT(\tau) & =cT_{0}+\frac{c^{2}}{\alpha}\left\{ \sinh\left[\tanh^{-1}\left(\frac{u_{0}}{c}\right)+\frac{\alpha\tau}{c}\right]-\frac{u_{0}\gamma_{0}}{c}\right\} \end{align}