Show that the function $f(z)=frac{1-cos z}{z^n}$ has no anti-derivative on $mathbb Csetminus{0}$ if $n$ is an odd integer.

Show that the function $f(z)=frac{1-cos z}{z^n}$ has no anti-derivative on $mathbb Csetminus{0}$ if $n$ is a positive odd integer.

I think, If we can able to prove $int_S f(z) dz neq 0$, on the unit circle $S$, then by fundamental theorem of algebra gives the desired one. But, how can I use the definition $int_gamma f(z)dz=int_a^b f(gamma(t))gamma'(t)dt$ for a continuous function $f(z)$ on a piecewise smooth contour $gamma:=gamma(t),~a leq gamma leq b$ ?