Compute $\oint_{S^1}\frac{z}{\sin^{3}(z/2)}dz$ without using Laurent series

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Let $D=\{z\in\mathbb{C}:|z|<1 \}$. How to compute the integral : $$\int_{+\partial D}\frac{z}{\sin^{3}(z/2)}dz$$ without using Laurent series?

My trouble is that I cannot find the poles and the residues.