= Solution
Where $\mathbf A=|\mathbf v|^2\mathbf w$ is twice continuously differentiable, the identity <divergence of a curl is zero> gives $\nabla\cdot(\nabla\times\mathbf A)=0$. All regular portions of the parameter interval lie in this region, hence
$$
\boxed{\int_{\Gamma_2}\nabla\cdot[\nabla\times(|\mathbf v|^2\mathbf w)]\,ds=0}
$$
as an <improper line integral>. Besides the $\operatorname{arctanh}(x^2)$ endpoint problem from (i), the <curve> touches $y=1$ at $\theta=\pi/2$, where the printed real $\arcsin y$ expression has no smooth open neighbourhood. The identity applies on the intervening open segments, and its zero value has a unique continuous extension along the <curve>. A literal classical integrand at every parameter is not supplied by the printed field.
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