How to integrate 1/( sin x + cos x)^2

 

\begin{array}{l} \smallint \frac{{\bf{1}}}{{{{\left( {sinx + cosx} \right)}^{\bf{2}}}}}{\rm{d}}x\\\\ \smallint \frac{1}{{{{\sin }^2}x + 2\sin x\cos x + {{\cos }^2}x}}{\rm{d}}x\\\\ \smallint \frac{1}{{1 + \sin 2x}}{\rm{d}}x\\\\ \smallint \frac{1}{{1 + \sin 2x}} \cdot \frac{{1 - \sin 2x}}{{1 - \sin 2x}}{\rm{d}}x\\\\ \smallint \frac{{1 - \sin 2x}}{{1 - {{\sin }^2}2x}}{\rm{d}}x\\\\ \smallint \frac{1}{{{{\cos }^2}2x}} - \frac{{\sin 2x}}{{{{\cos }^2}2x}}{\rm{d}}x\\\\ \smallint {\sec ^2}2x\; - \tan 2x\sec 2x{\rm{d}}x\\\\ \frac{{\tan 2x}}{2} - \frac{{\sec 2x}}{2} + c\\ \end{array}