Answer
$\frac{1}{24}(2\cos^33x\sin 3x + 9x + 3\sin 3x\cos 3x) + C$
Work Step by Step
Integrate by tables
$\int cos^43xdx$
Let $u=3x, du=3dx, n=4$
Rewrite
$\frac{1}{3} \int cos^nudu$
Use Formula #51 from the tables
$\frac{1}{3}[\frac{cos^33xsin3x}{4} + \frac{3}{4} \int cos^2udu]$, Integrate using Formula #49
$\frac{1}{3}[\frac{cos^33xsin3x}{4} + \frac{3}{4}[\frac{1}{2}(3x + sin3xcos3x)]] +C$
$\frac{1}{3}(\frac{cos^33xsin3x}{4} + \frac{9}{8}x + \frac{3}{8}sin3xcos3x)+C$
$\frac{1}{24}(2cos^33xsin3x + 9x + 3sin3xcos3x) + C$
