Solve an integral $\int\frac{\cos^3 x}{\sin^3 x+\cos^3 x}dx$ Solve an integral $$\int\frac{\cos^3 x}{\sin^3 x+\cos^3 x}dx$$
I tried to divide the numerator and denominator by $\cos^4 x$ to get $\sec x$ function but the term ${\sin^3 x}/{\cos^4 x}$ gives $\tan^2 x\sec^2 x\sin x$. How to get rid of $\sin x$ term?

$\displaystyle\frac{\cos^3x}{\sin^3x+\cos^3x}=\frac{1}{\tan^3x+1}=\frac{\tan^2x+1}{(\tan^2x+1)(\tan^3x+1)}=\frac{1}{\cos^2x(\tan^2x+1)(\tan^3x+1)}$
Now replace $\displaystyle t=\tan x,dt=\frac{dx}{\cos^2x}$ and do partial fractions.
