Official solution
Let, I = \[\int {\sqrt {1 + 2\cot x\cos ec + 2{{\cot }^2}x} .dx} \]
\[ \Rightarrow \] I = \[\int {\sqrt {{{{{\sin }^2}x + 2\cos x + 2{{\cos }^2}x} \over {{{\sin }^2}x}}} .dx} \]
\[ \Rightarrow \] I = \[\int {\sqrt {{{1 + 2\cos x + {{\cos }^2}x} \over {\sin x}}} .dx} \]
\[ \Rightarrow \] I = \[\int {\left| {{{1 + \cos x} \over {\sin x}}} \right|dx} \]
\[ \Rightarrow \] I = \[\int {\left| {\cos ec\,x + \cot x} \right|.dx} \]
\[ \Rightarrow \] I = \[\log \left| {\cos ec\,x - \cot x} \right| + \log \left| {\sin x} \right| + C\]
\[ \Rightarrow \] I = \[\log \left| {1 - \cos x} \right| + C\]
\[ \Rightarrow \] I = \[\log \left| {2{{\sin }^2}{x \over 2}} \right| + C\]
\[ \Rightarrow \] I = \[\log \left| {{{\sin }^2}{x \over 2}} \right| + \log 2+ C\]
\[ \Rightarrow \] I = 2\[\log \left| {{{\sin }}{x \over 2}} \right| + C_1\]