integration

\hspace{-16}\int\frac{1}{\sin x+\cos x+\tan x+\cot x+\sec x+\csc x}dx\\\\\\ $Now Put $\tan \left(\frac{x}{2}\right)=t\Leftrightarrow dx=\frac{2dt}{1+t^2}$\\\\ So $\sin x=\frac{2t}{1+t^2}\;$ and $\cos x=\frac{1-t^2}{1+t^2}dt$ and $\tan x=\frac{2t}{1-t^2}$\\\\\\ So $\int\frac{1}{\frac{2t}{1+t^2}+\frac{1-t^2}{1+t^2}+\frac{2t}{1-t^2}+\frac{1-t^2}{2t}+\frac{1+t^2}{1-t^2}+\frac{1+t^2}{2t}}.\frac{2dt}{(1+t^2)}$\\\\\\ $\int\frac{1}{\frac{1-t^2+2t}{1+t^2}+\frac{1+t^2+2t}{1-t^2}+\frac{1}{t}}\times \frac{2t}{(1+t^2)}dt$\\\\\\ $\int\frac{2t.(1-t)}{2t^4-2t^3+3t^2+t+1}dt$