I'm working with the system of differential equations: $$\begin{align*} \left\{ \begin{array} { l l } { (u')^2 + (v')^2 = 1 } \\ {u'v'' - u''v' = -v' + u' } \end{array} \right. \end{align*}$$ Where $u, v:I \to \mathbb{R}$. Now, with a little bit of work, one can show that (below $c$ is a constant): $$u'(s) = \frac { 2 \sqrt { 2 } \operatorname { tanh } \left[ \frac { s } { \sqrt { 2 }}+c \right] - 2 \operatorname { tanh } \left[ \frac { s } { \sqrt { 2 } } +c\right ] ^ { 2 } } { 2 - 2 \sqrt { 2 } \tanh \left[ \frac { s } { \sqrt { 2 } } +c\right] + 2 \tanh \left[ \frac { s } { \sqrt { 2 } }+c \right] ^ { 2 } }$$
And if there was a nice primitive to this function, my life would be easier, since $v = \pm \int \sqrt{1-(u')^2} \ ds$. But unfortunately there isn't. My ultimate goal here is to plot the curve: $$X(s) = (\cos(u(s)),\sin(u(s)), v(s) )$$ But due to how nasty all these expressions become, it's proven to be a pretty hard task. Naturally I think the only way for me to really "see" this curve is using numerical resources, but I'm new to Mathematica (so I'm sorry if this question is too basic, I wouldn't know), so I don't know how to do that and I'd really appreciate some help.