How to evaluate $\int\sin ^3 x\cos^3 x\:dx$ without a reduction formula?

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

You may write $$\sin^3 x \cos^3 x= \frac{1}{8} \sin^3(2 x)= \frac{1}{32}(3 \sin 2x - \sin 6x)$$ that is easy to integrate.