$$\int{\frac{c\sin x+d\cos x}{a\sin x+b\cos x}}dx$$

$$= \int{\frac{A(a\cos x-b\sin x)+B(a\sin x+b\cos x)}{a\sin x+b\cos x}}dx$$

$$= \int{\frac{A(a\cos x-b\sin x)}{a\sin x+b\cos x}}dx+\int{\frac{B(a\sin x+b\cos x)}{a\sin x+b\cos x}}dx$$

$$=A \int{\frac{1}{a\sin x+b\cos x}}d(a\sin x+b\cos x)+\int{B}dx$$

$$= A\ln{|a\sin x+b\cos x|}+Bx+C$$

其中a,b,c,d为任意常数,A,B为待定系数,联立两个式子即可获得