I am aware of the theorem that a function in the complex plane $f(z)$ has no antiderivative if and only if the contour integral of $f(z)$ over every closed path is non-zero. Because it is also a theorem that each closed path is contractable to a circle, therefore this is the same as integrating it over ever possible circle.
However, the integral is zero (so it's supposed to have an anti-derivative) whenever the circle is centered at the origin. But I know that this function never has an antiderivative. So what went awry?