$3 \cos x+3 \sin x=a(4 \sin x+5 \cos x)+b(d / d x(4 \sin x+5 \cos x)$
$3 \cos x+3 \sin x=\cos x(5 a+4 b)+\sin x(4 a-5 b)$
Compare the coefficients of sin $x$ and $\cos x$ on the both sides
$(5 a+4 b)=3,(4 a-5 b)=3$
$a=27 / 41, b=-3 / 41$
$\int \frac{3 \cos x+3 \sin x}{4 \sin x+5 \cos x} d x=\int \frac{27}{41} d x-\frac{3}{41} \int \frac{4 \cos x-5 \sin x}{4 \sin x+5 \cos x} d x$
$\int \frac{3 \cos x+3 \sin x}{4 \sin x+5 \cos x} d x=\frac{27}{41} x-\frac{3}{41} \ln |4 \sin x+5 \cos x|+c$
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