$\mathrm{I}=\underset{\mathrm{\pi }/8}{\overset{3\mathrm{\pi }/8}{\int }}\frac{11+4\cos 2\mathrm{x}+\cos 4\mathrm{x}}{1-\cos 4\mathrm{x}}\mathrm{dx}$          .....(1)

Using King

$\mathrm{I}=\underset{\mathrm{\pi }/8}{\overset{3\mathrm{\pi }/8}{\int }}\frac{11-4\cos 2\mathrm{x}+\cos 4\mathrm{x}}{1-\cos 4\mathrm{x}}\mathrm{dx}$         .....(2)

(1) + (2)

$\text{2I=2}\underset{\pi /8}{\overset{3\pi /8}{\int }}\frac{11+cos4x}{1-cos4x}\hspace{0.17em}dx$

$\Rightarrow \mathrm{I}=\underset{\mathrm{\pi }/8}{\overset{3\mathrm{\pi }/8}{\int }}\frac{12-(1-\cos 4\mathrm{x})}{1-\cos 4\mathrm{x}}\mathrm{dx}=12\underset{\mathrm{\pi }/8}{\overset{3\mathrm{\pi }/8}{\int }}\frac{1}{2{\sin }^{2}2\mathrm{x}}\mathrm{dx}-\underset{\mathrm{\pi }/8}{\overset{3\mathrm{\pi }/8}{\int }}\mathrm{dx}$

$=6\underset{\mathrm{\pi }/8}{\overset{3\mathrm{\pi }/8}{\int }}{\mathrm{cosec}}^{2}2\mathrm{x}\mathrm{dx}-\frac{\mathrm{\pi }}{4}$

$=-{\frac{6}{2}\cot 2\mathrm{x}]}_{\frac{3\mathrm{\pi }}{8}}^{\frac{\mathrm{\pi }}{8}}-\frac{\mathrm{\pi }}{4}=-3[\left(1\right)+\left(1\right)]-\frac{\mathrm{\pi }}{4}=-6-\frac{\mathrm{\pi }}{4}$

$=-6-\frac{\mathrm{\pi }}{4}$