$${tg}\left(\frac{{x}}{\mathrm{2}}\right)={t} \\$$$${dx}=\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} } \\$$$$\Rightarrow{sin}\left({x}\right)=\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\:\:{cos}\left({x}\right)=\frac{\mathrm{1}−{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} } \\$$$$\Rightarrow\int\frac{{dx}}{\mathrm{5}{sin}\left({x}\right)−\mathrm{4}{cos}\left({x}\right)+\mathrm{3}}=\int\frac{\mathrm{2}{dt}}{\mathrm{3}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)+\mathrm{10}{t}−\mathrm{4}+\mathrm{4}{t}^{\mathrm{2}} } \\$$$$=\int\frac{{dt}}{\mathrm{7}{t}^{\mathrm{2}} +\mathrm{10}{t}−\mathrm{1}}=\int\frac{{dt}}{\mathrm{7}\left({t}+\mathrm{5}−\sqrt{\mathrm{32}}\right)\left({t}+\mathrm{5}+\sqrt{\mathrm{32}}\right)} \\$$$$=\int\left[\frac{\mathrm{1}}{\mathrm{14}\sqrt{\mathrm{32}}\left({t}+\mathrm{5}−\sqrt{\mathrm{32}}\right)}−\frac{\mathrm{1}}{\mathrm{14}\sqrt{\mathrm{32}}\left({t}+\mathrm{5}+\sqrt{\left.\mathrm{32}\right)}\right.}\right]{dt} \\$$$$=\frac{\mathrm{1}}{\mathrm{14}\sqrt{\mathrm{32}}}{ln}\mid\frac{{t}+\mathrm{5}−\sqrt{\mathrm{32}}}{{t}+\mathrm{5}+\sqrt{\mathrm{32}}}\mid+{c} \\$$$$\\$$$$\Rightarrow\int\frac{{dx}}{\mathrm{5}{sin}\left({x}\right)−\mathrm{4}{cos}\left({x}\right)+\mathrm{3}}=\frac{\mathrm{1}}{\mathrm{14}\sqrt{\mathrm{32}}}{ln}\mid\frac{{tan}\left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{5}−\sqrt{\mathrm{32}}}{{tg}\left(\frac{{x}}{\mathrm{2}}\right)+\mathrm{5}+\sqrt{\mathrm{32}}}\mid+{c} \\$$$$\\$$