Answer:
Let \[l=\int{{{\{1+2\tan \,x(\tan \,x+\sec \,x)\}}^{1/2}}dx}\] \[=\int{{{\{1+2{{\tan }^{2}}x+2\tan x\sec \,x\}}^{1/2}}dx}\]\[=\int{{{\{1+{{\tan }^{2}}x+{{\tan }^{2}}x+2\tan x\sec x\}}^{1/2}}dx}\] \[=\int{{{\{{{\sec }^{2}}x+{{\tan }^{2}}x+2\tan x\sec x\}}^{1/2}}dx}\] \[=\int{{{\{{{(\sec x+\tan x)}^{2}}\}}^{1/2}}dx}\] \[=\int{(\sec \,x+\tan x)\,dx}\] \[=\log |\sec x+\tan x|+\log |\sec x|+\,C\]
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