calculus - How to evaluate $intsin ^3 xcos^3 x:dx$ without a reduction formula?

We have the integral $$\displaystyle\int \sin ^3 x \cos^3 x \:dx.$$ You can do this using the reduction formula, but I wonder if there's another (perhaps simpler) way to do this, like for example with a substitution?

Answer

Hint. You may write
$$
\sin^3 x \cos^3 x= \sin x(1 - \cos^2x)\cos^3 x=\sin x(\cos^3x - \cos^5x)
$$