`int{1 + 2 tan x(tan x + sec x)}^(1/2)dx`

Evaluate: int{1+2tan x(tan x+sec x))^((1)/(2))dx

If int[1+2tan x(tan x+sec x)]^(1/2)dx=lambda log|(tan((x)/(2)+k))/(cos x)|+C then lambda K is equal to where 0 < x < (pi)/(2)

intsqrt(1+2 tan x (tan x + sec x ) ) dx

int sqrt(1+2tan x(tan x+sec x))dx is equal to

int (tan^(2)x)/(sec x -1)dx

int(sec x+tan x)^(2)dx

int(sec x)/(tan^(2)x)dx

int(tan^(2)x)/(sec x-1)dx

int(tan x+sec x-1)/(tan x-sec x+1)dx