Finding a unit speed parametrisation for $\alpha(t)= 0.5 (t, 1/t, \sqrt{2} \log(t))$

Question

I'm trying to find a unit speed parametrisation for the curve $\alpha: (0, \infty) \to \mathbb{R}^3$ s.t $$\alpha(t)= 0.5 (t, 1/t, \sqrt{2} \log(t)).$$

However, $$s(t) = 0.5 (t - \frac{ 1}{ t} ),$$ and as $t \to 0$, $s \to \infty $, so I don't know how to interpret this result, or how to find $t(s)$.

I also tried reparametrization of the given curve with $x \to e^x$, but that gave me $$s = \int_{-\infty}^\infty (e^x)*(3 + e^{-4x})^{0.5}dx,$$ but this integral also does not converge.

Question:

How to find a unit speed parametrisation for such a curve ?

Answer 1

Solve for $t$:

$$t-\frac1t=2s$$ or $$t^2-2st-1=0,$$ which gives $$t=s\pm\sqrt{s^2+1}.$$