Question Number 207424 by MetaLahor1999 last updated on 14/May/24
$${f}_{{n}} \left({x}\right):=\int{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \frac{{cos}\left({x}\right)}{\:\left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{{n}}{\mathrm{3}}} }{dx}=…? \\ $$$${for}\:{n}=\mathrm{1},\:{i}\:{found}\: \\ $$$$\:\:\:\:\:\:{f}_{\mathrm{1}} \left({x}\right)=\frac{\mathrm{3}}{\mathrm{4}}{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\:{C} \\ $$$${is}\:{there}\:{any}\:{ideas}\:{for}\:{a}\:{general}\:{case}\:{or} \\ $$$${the}\:{case}\:{n}=\mathrm{2}? \\ $$