Answer
$-\dfrac{\sin2\theta\cos^42\theta}{10}+\dfrac{\sin2\theta\cos^22\theta}{30}+\dfrac{\sin2\theta}{15}+C$
Work Step by Step
Apply the Reduction Formula $\int\sin^nax\cos^maxdx=-\dfrac{\sin^{n-1}ax\cos^{m+1}ax}{a(m+n)}+\frac{n-1}{m+n}\int\sin^{n-2}ax\cos^m \space (ax) dx$ and $\int\sin^nax\cos^maxdx=\frac{\sin^{n+1}ax\cos^{m-1}ax}{a(m+n)}+\frac{m-1}{m+n}\int\sin^{n}ax\cos^{m-2}axdx$
Let $I=-\dfrac{\sin2\theta\cos^42\theta}{10}+\dfrac{1}{5}\int\cos^32\theta d\theta \\=-\dfrac{\sin2\theta\cos^42\theta}{10}+\dfrac{1}{5}(\dfrac{\sin2\theta\cos^22\theta}{6}+\dfrac{2}{3} \times \int\cos2\theta d\theta) \\=-\dfrac{\sin2\theta\cos^42\theta}{10}+\dfrac{1}{5}(\dfrac{\sin2\theta\cos^22\theta}{6}+\dfrac{2}{3}\times\dfrac{\sin2\theta}{2})+C \\=-\dfrac{\sin2\theta\cos^42\theta}{10}+\dfrac{\sin2\theta\cos^22\theta}{30}+\dfrac{\sin2\theta}{15}+C$
