$$\begin{aligned} & I=\int(\operatorname{cosec} x)^5 d x=\int(\operatorname{cosec} x)^3(\operatorname{cosec} x)^2 d x \\ & =(\operatorname{cosec} x)^3 \int \operatorname{cosec}^2 x d x- \\ & \int\left(\frac{d}{d x}(\operatorname{cosec} x)^3 \int \operatorname{cosec}^2 x d x\right) d x \end{aligned}$$

$$\begin{aligned} & =-\cot x(\operatorname{cosec} x)^3-\int 3 \operatorname{cosec}^2 x \cdot(-\operatorname{cosec} x \cot x)(-\cot x) d x \\ & =-\cot x(\operatorname{cosec} x)^3-\int 3 \operatorname{cosec}^3 x \cot ^2 x d x \\ & =-\cot x(\operatorname{cosec} x)^3-3 \int(\operatorname{cosec} x)^3\left(\operatorname{cosec}^2 x-1\right) d x \\ & I=-\cot x(\operatorname{cosec} x)^3-3 I+3 \int(\operatorname{cosec} x)^3 d x \\ & 4 I=-\cot x(\operatorname{cosec} x)^3+3 \int(\operatorname{cosec} x)^3 d x \\ & =-\cot x(\operatorname{cosec} x)^3+3 I_1 \\ & I_1=\int \operatorname{cosec} x \cdot \operatorname{cosec}^2 x d x=\operatorname{cosec} x(-\cot x)- \\ & \qquad \int(-\operatorname{cosec} x \cot x)(-\cot x) d x \end{aligned}$$

$$\begin{aligned} & I_1=-\operatorname{cosec} x \cot x-\int \operatorname{cosec} x\left(\operatorname{cosec}^2 x-1\right) d x \\ & I_1=-\operatorname{cosec} x \cot x-I+\int \operatorname{cosec} x d x \\ & 2 l_1=-\operatorname{cosec} x \cot x+\ln \left|\tan \frac{x}{2}\right| \\ & \Rightarrow \quad 4 l=-\cot x(\operatorname{cosec} x)^3-\frac{3}{2} \operatorname{cosec} x \cot x +\frac{3}{2} \ln \left|\tan \frac{x}{2}\right|+C \end{aligned}$$

$$\begin{aligned} & I=-\frac{1}{4} \operatorname{cosec} x \cot x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\frac{3}{8} \ln \left|\tan \frac{x}{2}\right|+C \\ & \Rightarrow \alpha=-\frac{1}{4}, \beta=\frac{3}{8} \Rightarrow 8(\alpha+\beta)=1 \end{aligned}$$