Start by using the Pythagorean identity to replace cos4 with (1- sin2)2 = (1-2 sin2 + sin4) .
The integrand is now
sin4(1- 2 sin2 + sin4) cos = ( sin4 - 2 sin6 + sin8) cos
This can be integrated term by term using the u substitution u = sin.
However, it is easier to notice the cos factor is an integrating factor for each term.
The antiderivative can be written down right away. It is
F = (1/5) sin5 - (2/7) sin7 + (1/9) sin9 + k