Let $f(u,v) = u + iv$ where $u = \cosh{x}\cos{y}$ and $v=\sinh{x}\sin{y}$
Note that
\begin{equation*}
\begin{aligned}
\frac{\partial u}{\partial x} &= \sinh{x}\cos{y}\\
\frac{\partial v}{\partial y} &= \sinh{x}\cos{y}\\
\frac{\partial u}{\partial y} &= -\cosh{x}\sin{y}\\
\frac{\partial v}{\partial x} &= \cosh{x}\sin{y}
\end{aligned}
\end{equation*}
Since
\begin{equation*}
\begin{aligned}
\frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} \\
\frac{\partial u}{\partial y} &\neq -\frac{\partial v}{\partial x}
\end{aligned}
\end{equation*}
We know that $f$ is not analytic on $D$ by the contrapositve of Cauchy-Riemann Theorem.
So, $f$ is not sourceless or irrotational on $D$.
The error you are talking about is that the textbook used $dx$ instead of $\partial x$
Thus, the proper equations should be
\begin{equation*}
\begin{aligned}
\int_{\gamma} u dx+v dy &= \iint_{\Omega} \left[\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right]dxdy=0 \\
\int_{\gamma} u dy - v dx &= \iint_{\Omega} \left[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right]dxdy=2 \text{ area}(\Omega)
\end{aligned}
\end{equation*}
