$A:\int \frac{2\mathrm{cosx}+3sinx}{4\cos x+5\sin x}dx=$$\frac{-2}{41}\log |4\cos x+5\sin x|+\frac{23}{41}x+c$

$R:\int \frac{a\cos x+b\sin x}{c\cos x+d\sin x}dx=$$\frac{ac+bd}{c^2+d^2}x+\frac{ad-bc}{c^2+d^2}\log |c\cos x+d\sin x|+k$

$$\begin{gathered} R:\int \frac{a\mathrm{cosx}+bsinx}{C\cos x+d\sin x}dx \\ =\frac{ac+bd}{c^2+d^2}x+\frac{ad-bc}{c^2+d^2}\log \left|c \cos x+d \sin x\right|+C \\ is correct \\ A:\int \frac{2\cos x+3 \sin x}{4\cos x+5\sin x}dx \\ =\left(\frac{2\left(4\right)+3\left(5\right)}{4^2+5^2}\right)x+\frac{2\left(5\right)-4\left(3\right)}{4^2+5^2}\log \left|4 \cos x+5\sin x\right|+C \\ =\frac{23}{41}x-\frac{2}{41}\log \left|4\cos x+5\sin x\right|+C \end{gathered}$$

Both A and R are true and R is the correct  explanation of A