Solution:

Given that,
$\int \frac{\sqrt{1+\cos 2 x}}{\sqrt{1-\cos 2 x}} d x$
It is known that
$\begin{array}{l}
1+\cos 2 x=2 \cos ^{2} x \\
1-\cos 2 x=2 \sin ^{2} x
\end{array}$
On substituting these formulae in the equation given we obtain
$\Rightarrow \int \sqrt{\frac{2 \cos ^{2} x}{2 \sin ^{2} x}} d x$
By applying standard formula, we obtain
$\Rightarrow \int \sqrt{\cot ^{2} x} d x$
By simplifying we obtain
$\begin{array}{l}
\Rightarrow \int \cot x d x \\
\Rightarrow \log |\sin x|+c
\end{array}$