$\int \{1+2\tan x(\tan x+\sec x){\}}^{1/2}dx$   is equal 

Solution:

 We have, 

$\begin{array}{l}\Rightarrow I=\int \{1+2\tan x(\tan x+\sec x){\}}^{1/2}dx\\ \Rightarrow I=\int {\left|1+{\tan }^{2}x+{\tan }^{2}x+2\tan x\sec x\right|}^{1/2}dx\\ \Rightarrow I=\int {\left({\sec }^{2}x+{\tan }^{2}x+2\tan x\sec x\right)}^{1/2}dx\end{array}$

$\begin{array}{l}\Rightarrow I\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\int (\tan x+\sec x)dx\\ \Rightarrow I\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\log \sec x+\log (\sec x+\tan x)+C\\ \Rightarrow I\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\log \sec x(\sec x+\tan x)+C\end{array}$