Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$ \alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t). $$ Show that $$ \int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz. $$
$1/z$ is holomorphic on $\mathbb{C}\setminus\left\{0\right\}$. Both ways are closed ways with start- and endpoint $a$. So to my opinion all conditions of the Cauchy integral theorem are fullfilled clearly except that both ways are homotopic.
I am thinking of $$ H\colon [0,1]\times [0,1]\to\mathbb{C}, H(x,y):=a\cos(2\pi x)+i(1-y)a\sin(2\pi x)+iyb\sin(2\pi x) $$ as an appropriate homotopic function.
Anyhow it is $$ H(x,0)=\alpha(x),~~~~~H(x,1)=\beta(x). $$