Integrating $\int \cos^3(x)\cos(2x) \, dx$ How would Integrate the following?
$$\int \cos^3(x)\cos(2x) \, dx.$$
I did 
$$\int \cos^3(x)(1-2\sin^2(x)) \, dx = 2\int \cos^3(x)-\cos^3x\sin^2x \, dx$$
But I find myself stuck....

\begin{align}
\int \cos^3(x) \cos(2x) dx &= \int \cos^3(x)(1-2\sin^2(x)) dx \\
&= \int \cos^2(x) \cdot \cos(x) (1-2\sin^2(x)) dx \\
&= \int (1-\sin^2x)\cdot \cos(x) \cdot(1-2\sin^2(x)) dx \\
&= \int ( \cos(x)- 3 \sin^2x \cdot \cos(x) + 2 \sin^4x \cdot \cos(x) ) dx.
\end{align}
Now using inverse chain rule which is 
$$\int f^n(x) \cdot f'(x) dx= f^{n+1}(x) +C$$
then
$$ \int \cos^3(x) \cos(2x) dx  = \sin(x)- \sin^3x + \frac{2}{5} \, \sin^5x + C$$
