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

 

$$\begin{array}{l}\int \frac{\mathbf{1}}{{\left(sinx+cosx\right)}^{\mathbf{2}}}\mathrm{d}x\\ \\ \int \frac{1}{{\sin }^{2}x+2\sin x\cos x+{\cos }^{2}x}\mathrm{d}x\\ \\ \int \frac{1}{1+\sin 2x}\mathrm{d}x\\ \\ \int \frac{1}{1+\sin 2x}\cdot \frac{1-\sin 2x}{1-\sin 2x}\mathrm{d}x\\ \\ \int \frac{1-\sin 2x}{1-{\sin }^{2}2x}\mathrm{d}x\\ \\ \int \frac{1}{{\cos }^{2}2x}-\frac{\sin 2x}{{\cos }^{2}2x}\mathrm{d}x\\ \\ \int {\sec }^{2}2x-\tan 2x\sec 2x\mathrm{d}x\\ \\ \frac{\tan 2x}{2}-\frac{\sec 2x}{2}+c\end{array}$$