Answer
$-\frac{1}{24(1+2\sin 4\theta)^3}+C$
Work Step by Step
Let $u=1+2\sin4\theta$. Then $du=8\cos 4\theta d\theta$, so $\cos4\theta d\theta=\frac{1}{8}du$.
$\int \frac{\cos 4\theta}{(1+2\sin4\theta)^4}d\theta$
$=\int \frac{\frac{1}{8}du}{u^4}$
$=\frac{1}{8}\int u^{-4}du$
$=\frac{1}{8}*\frac{u^{-3}}{-3}+C$
$=-\frac{1}{24u^3}+C$
$=-\frac{1}{24(1+2\sin 4\theta)^3}+C$
