Mathematics - Indefinite Integration Question with Solution | TestHub

$\frac{1}{2\sqrt{2}}\left[2{\tan }^{-1}\left(\frac{\tan x-1}{\sqrt{2\tan x}}\right)+\log \left[\frac{\tan x-\sqrt{2\tan x}+1}{\tan x+\sqrt{2\tan x}+1}\right]\right]+c$

$\frac{1}{2\sqrt{2}}\left[2{\tan }^{-1}\left(\frac{\tan x-1}{2\tan x}\right)+\log \left[\frac{\tan x-\sqrt{2\tan x}+1}{\tan x+\sqrt{2\tan x}+1}\right]\right]+c$

$\frac{1}{2\sqrt{2}}\left[2{\tan }^{-1}\left(\frac{\tan x+1}{\sqrt{2}\tan x}\right)+\log \left[\frac{{\tan }^{2}x-2\tan x+1}{\tan x+2\tan x+1}\right]\right]+c$

$\frac{1}{5\sqrt{2}}\left[2{\tan }^{-1}\left(\frac{\tan x+1}{\sqrt{2}\tan x}\right)+\log \left[\frac{{\tan }^{2}x-2\tan x-1}{\tan x+2\tan x+1}\right]\right]+c$