$\int \frac{\cos ^3 x d x}{\sin x+\cos x}$

\begin{aligned} & \frac{\cos ^3 x}{\sin x+\cos x} \\ & =\frac{\cos x\left(1-\sin ^2 x\right)}{\sin x+\cos x} \\ & =\frac{\cos x}{\sin x+\cos x}-\frac{\cos x \sin ^2 x}{\sin x+\cos x} \end{aligned}

\begin{aligned} & A=\int \frac{\cos x}{\sin x+\cos x} d x \\ & B=\int \frac{\cos x \sin ^2 x}{\sin x+\cos x} d x \end{aligned}

\begin{aligned} & A=\int \frac{\cos x}{\sin x+\cos x} d x \\ & =\int \frac{1}{2}\left(\frac{\cos x-\sin x}{\sin x+\cos x}+\frac{\sin x+\cos x}{\sin x+\cos x}\right) d x \\ & =\frac{1}{2} \int \frac{d(\sin x+\cos x)}{\sin x+\cos x}+\frac{1}{2} \int d x \\ & =\frac{1}{2} \ln |\sin x+\cos x|+\frac{x}{2}+C_1 \end{aligned}
\begin{aligned} & B=\int \frac{\cos x \sin ^2 x}{\sin x+\cos x} d x \\ & =-\int \frac{\sin x \cos x}{\sin x+\cos x} d \cos x \\ & =-\int \frac{\sin x \cos x d \cos x}{\sin x+\cos x}-\frac{1}{2} \int \frac{\left(\sin ^2 x+\cos ^2 x\right) d \cos x}{\sin x+\cos x}+\frac{1}{2} \int \frac{d \cos x}{\sin x+\cos x} \\ & =-\frac{1}{2} \int \frac{(\sin x+\cos x)^2 d \cos x}{\sin x+\cos x}-\frac{1}{2} \int \frac{\sin x d x}{\sin x+\cos x} \\ & =-\frac{1}{2} \int \sin x d \cos x-\frac{1}{2} \int \cos x d \cos x-\frac{1}{2} \int \frac{\sin x d x}{\sin x+\cos x} \\ & =\frac{1}{2} \int\left(\frac{1-\cos 2 x}{2}\right) d x-\frac{1}{4}(\cos x)^2-\frac{1}{2} \int \frac{\sin x d x}{\sin x+\cos x} \\ & =\frac{1}{4}\left(x-\frac{\sin 2 x}{2}\right)-\frac{1}{4} \cos ^2 x-\frac{1}{2}\left(\frac{x}{2}-\frac{1}{2} \ln |\sin x+\cos x|\right)+C_2 \end{aligned}

(核心依旧是降次，将 $\cos x \sin ^2 x$ 三次项降为二次项，以上做法有一定的技巧性但不难观察出 来。)

\begin{aligned} & \int \frac{\cos ^3 x d x}{\sin x+\cos x}=A-B \\ & =\frac{x}{2}+\frac{\sin 2 x}{8}+\frac{1}{4} \cos ^2 x+\frac{1}{4} \ln |\sin x+\cos x|+C \end{aligned}
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