Use Cauchy Residue Theorem to find the integral

Homework Statement

To find the integral by Cauchy Residue Theorem and apply substitution method.

Homework Equations

To show: ∫$^{2∏}_{0}$$\frac{cosθ}{13+12cosθ}$=-$\frac{4∏}{15}$

The Attempt at a Solution

The solution I have done is attached. It is different as what the question wants me to show. I do not know where I did it wrong.

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$$6z^3+13z^2+6z = 6 z(z+2/3)(z+3/2) \neq z(z+2/3)(z+3/2)$$