Integrate : $\int \large\frac{\sin x +\cos x }{\sqrt {\sin 2x}}$$dx$

Question

$(a)\;\frac{-1}{2}\tan ^{-1} (\sqrt {\frac{\tan x }{2}})+c \\(b)\;\frac{1}{2}\tan ^{-1} (\tan x)+c \\(c)\;\frac{1}{2}\tan ^{-1} (\sqrt {\frac{-1}{\sqrt {2 \tan x }}})+c \\ (d)\;\frac{1}{2}\tan ^{-1} (\sqrt {\frac{\tan x }{2}}-\frac{1}{\sqrt {2 \tan x }})+c$