Where the method used should be using complex analysis.

$${\int}_{c}\frac{d\theta}{(p+\mathrm{cos}\theta {)}^{2}}=\frac{2\pi p}{({p}^{2}-1)\sqrt{{p}^{2}-1}};c:\left|z\right|=1$$

$${\int}_{c}\frac{d\theta}{(p+\mathrm{cos}\theta {)}^{2}}=\frac{2\pi p}{({p}^{2}-1)\sqrt{{p}^{2}-1}};c:\left|z\right|=1$$