Fractional Trigonometric Integrands 
*

*$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$

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

*$$∫\frac{dx}{a\sin x+\cos x}$$
What are the relations between the numerator in the denominator, and what is the general pattern to solve these type of questions?

You can use the following way to calculate all of them once. Let
$$ A=\int \frac{\sin xdx}{d\sin x+e\cos x+f}, B=\int \frac{\cos xdx}{d\sin x+e\cos x+f}, C=\int \frac{dx}{d\sin x+e\cos x+f}. $$
It is easy to check
$$ dA+eB+fC=1, dB-eB=\ln|d\sin x+e\cos x+f|+Const. $$
So
$$ A=\frac{d(1-fC)}{d^2+e^2}-\frac{e\ln|d\sin x+e\cos x|}{d^2+e^2}+Const, B=\frac{e(1-fC)}{d^2+e^2}+\frac{f\ln|d\sin x+e\cos x|}{d^2+e^2}+Const.$$
Thus we only need to get $C$. Let
$$\tan \frac{x}{2}=t,x=2\arctan t. $$
Then
$$\sin x=\frac{2t}{1+t^2},\cos x=\frac{1-t^2}{1+t^2},dx=\frac{2}{1+t^2}dt.$$
Then we have
$$ C=\int \frac{dx}{d\sin x+e\cos x+f}=2\int\frac{dt}{(f-e)t^2+2dt+(f+e)}$$
which is not hard to get.
