$\int {\cfrac{{\cos 2x}}{{{{\left( {\sin x + \cos x} \right)}^2}}}} dx$ is equal to

Option b is correct

Let $I = \int {\cfrac{{\cos 2x}}{{{{\left( {\sin x + \cos x} \right)}^2}}}} dx = \int {\cfrac{{{{\cos }^2}x - {{\sin }^2}x}}{{{{\left( {\cos x - \sin x} \right)}^2}}}d} x$

$= \int {\cfrac{{\cos x - \sin x}}{{\cos x + \sin x}}} dx$
$= \log |\left( {\cos x + \sin x} \right)| + C$