Let $g\left( z\right) =e^{h\left( z\right) }$ ( you may not need the $ g $ expression.), $z\in \mathbb{D}$, $\beta :[0,b)\rightarrow \mathbb{D}$ curve, where $\mathbb{D}$ is a unitary disc in $\mathbb{C}$ and $h$ is holomorphic in $\mathbb{D}$.
Let $X:% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$ given by
$X\left( z\right) =\left(\Re \int_{0}^{z}\frac{1}{2}\left( \frac{1}{g\left( \zeta \right) }-g\left( \zeta \right) \right) d\zeta ,\Re\int_{0}^{z}\frac{i}{2}\left( \frac{1}{g\left( \zeta \right) }% +g\left( \zeta \right) \right) d\zeta ,\Re\int_{0}^{z}d\zeta \right)$
Show that $\left\Vert \left( X\circ \beta \right) ^{\prime }\left( t\right) \right\Vert =\frac{1}{\left\vert g\left( \beta \left( t\right) \right) \right\vert }+\left\vert g\left( \beta \left( t\right) \right) \right\vert $
For some time now I've been trying to check the equality, I've tried variable change, I used the $ g $ expression, but I got nowhere. Sounds easy, but I'm not stuffing.
Maybe I did not put all the hypotheses for the integral is well defined, but consider that yes.