Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve
Perhaps simpler is, assuming the domain of $\gamma $ is $[a,b]$,
$$\int _\gamma f=\int \limits _a^b f\left(\gamma (t)\right)\gamma '(t)\,\mathrm dt=\int \limits_a^b (F\circ \gamma)'(t)\,\mathrm dt=F\left(\gamma (b)\right)-F(\gamma(a))=0.$$