Let $\alpha (t) = a \exp(bt) (\cos t, \sin t)$, where $a > 0, b<0$.
We want to compute its arc length parameterization.
In order to do so, we compute the arc length,
$L_0^t \alpha = \int_0^t |\alpha'| = a \sqrt{1+b^2}[\frac{e^{bt}}{b}]_{0}^{t}$
The reparameterization desired is the inverse of the length, that is, $\phi(t) = \frac{1}{b} \ln (\frac{tb}{a\sqrt{1+b^2}}+1)$.
However, when I try to compute the tangent of $\beta = \alpha \circ\phi$ I get something which does not have constant module, let alone module 1.
What is going on? Is my reasoning incorrect, or am I just making a mistake in my calculations?