Find the length of curve given by: $$v=\ln{(u+\sqrt{a^2+u^2})}$$ which is situated on the helicoid surface: $$r(u,v)=[u\cos{v}, u\sin{v}, av]$$ and bounded by $v=0$ and $v=1$.
All I know is the formula for curve length: $$L=\int_a^b{\left|\frac{d}{dt}r[u(t), v(t)]\right|}dt$$
If we set $u=a \sinh(t)$, we have $v=\log(a)+t$ and the curve is given by $$\gamma(t)=\left(a\sinh(t)\cos(t+\log a),a\sinh(t)\sin(t+\log a),a(t+\log a)\right)$$ and $t$ ranges from $-\log(a)$ to $1-\log(a)$. Its arclength is so given by
$$ \int_{-\log a}^{1-\log a}\sqrt{2a^2\cosh^2(t)}\,dt =\color{red}{\frac{e-1}{e\sqrt{2}}(a^2+e)}.$$