Answer
$r(s)=(\dfrac{s}{\sqrt 3}+1) \lt 1, \sin (\ln (\dfrac{s}{\sqrt 3}+1)), \cos (\ln (\dfrac{s}{\sqrt 3}+1)) \gt$
Work Step by Step
Here, $r(t)=\lt e^t , e^t \sin t, e^t \cos t \gt =e^t\lt 1, \sin t, \cos t\gt ; |'(t)=e^t\lt 1, \cos t+\sin t, \cos t-\sin t\gt $
and $|r'(t)|^2=e^{2t}\lt 1, (\cos t+\sin t)^2, (\cos t-\sin t)^2 \gt=3e^{2t} $
or, $|r'(t)|=\sqrt 3e^{t} $
We need to solve for $t$
$e^t=s/\sqrt 3+1 \implies t=\ln (\dfrac{s}{\sqrt 3}+1)$
Now, plug this into $r(t)$ to get $r(s)$.
we have
$r(s)=(\dfrac{s}{\sqrt 3}+1) \lt 1, \sin (\ln (\dfrac{s}{\sqrt 3}+1)), \cos (\ln (\dfrac{s}{\sqrt 3}+1)) \gt$
