$\int \left(\frac{\cos x-1}{\sin x+1}\right)e^{x}dx=$

$\int \left[\frac{\cos x}{1+\sin x}+\frac{-1}{1+\sin x}\right]e^{x}dx$

$\begin{array}{l}\int e^x\left[f\left(x\right)+f^1\left(x\right)\right]dx=e^{x}f\left(x\right)+c\\ f\left(x\right)=\frac{\cos x}{1+\sin x}\end{array}$