$\begin{aligned} & f(x)=\int \operatorname{cosec}^5 x d x=\int \operatorname{cosec}^2 x \cdot \operatorname{cosec}^3 x d x \\ & =\operatorname{cosec}^3 x(-\cos x)-\int-\cot x \cdot 3 \operatorname{cosec}^2 x \\ & \quad \quad(-\operatorname{cosec} x \cot x) d x \\ & =-\cot x \operatorname{cosec}^3 x-3 \int \cot ^2 x \operatorname{cosec}^3 x d x \\ & =-\cot x \operatorname{cosec}^3 x-3 \int\left(\operatorname{cosec}^2 x-1\right) \operatorname{cosec}^3 x d x \\ & =-\cot x \operatorname{cosec}^3 x-3 \int \operatorname{cosec}^5 x d x+3 \int \operatorname{cosec}^3 x d x \\ & \quad f(x)=-\cot x \operatorname{cosec}^3 x-3 I+3 I_1 \\ & 4 f(x)=\cot x \operatorname{cosec}^3 x+3 I_1 \\ & \quad I_1=\int \operatorname{cosec}{ }^3 x d x \\ & =\int \operatorname{cosec} x \cdot \operatorname{cosec}^2 x d x\end{aligned}$
$\begin{aligned} & =\operatorname{cosec} x(-\cot x)-\int-\cot x \cdot(-\operatorname{cosec} x \cot x) d x \\ & =-\operatorname{cosec} x \cot x-\int \cot ^2 x \operatorname{cosec} x d x \\ & =-\operatorname{cosec} x \cot x-\int\left(\operatorname{cosec}^2 x-1\right) \operatorname{cosec} x d x \\ & =-\operatorname{cosec} x \cot x-\int \operatorname{cosec}^3 x d x+\int \operatorname{cosec} x d x \\ & 2 I_1=-\operatorname{cosec} x \cot x+\log (\operatorname{cosec} x-\cot x) \\ & f(x)=\frac{1}{4}\left[-\cot x \operatorname{cosec}^3 x+\frac{3}{2}(-\operatorname{cosec} x \cot x]\right. \\ & \quad+\log (\operatorname{cosec} x-\cot x)] \\ & f(\pi / 4)=\frac{1}{4}\left[-2 \sqrt{2}+\frac{3}{2}(-\sqrt{2}+\log (\sqrt{2}-1)]\right. \\ & =\frac{1}{8}[-4 \sqrt{2}-3 \sqrt{2}+3 \log (\sqrt{2}-1)] \\ & =-\frac{1}{8}[7 \sqrt{2}-3 \log (\sqrt{2}-1)] \\ & =-\frac{1}{8}[7 \sqrt{2}+3 \log (\sqrt{2}+1)]\end{aligned}$