How To Integrate 1+sin x /1+cos x

 

$$\begin{array}{l}\int \frac{\mathbf{1}+\mathbf{s}\mathbf{i}\mathbf{n}x}{\mathbf{1}+\mathbf{c}\mathbf{o}\mathbf{s}x}\mathrm{d}x\\ \\ \int \frac{1+\sin x}{1+\cos x}\ast \frac{1-cosx}{1-cosx}\mathrm{d}x\\ \int \frac{\left(1+\sin x\right)\left(1-\cos x\right)}{1-{\cos }^{2}x}\mathrm{d}x\\ \int \frac{1+\sin x-\cos x-\sin x\cos x}{{\sin }^{2}x}\mathrm{d}x\\ \int \frac{1}{{\sin }^{2}x}+\frac{\sin x}{{\sin }^{2}x}-\frac{\cos x}{{\sin }^{2}x}-\frac{\sin x\cos x}{{\sin }^{2}x}dx\\ \int {\csc }^{2}x+\csc x-\cot x\csc x-\cot x\mathrm{d}x\\ \Rightarrow -\cot x+\ln |\csc x-\cot x\left|+\csc x-{\ln }^{}|\sin x\right|+C\end{array}$$