TITLE: $ \int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx$
QUESTION [8 upvotes]: What is the value of
$$
\int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx
$$

I have no idea how to start with this integral, any hint?

REPLY [4 votes]: You may use the Laplace transform:
$$\mathcal{L}^{-1}\left(\frac{1}{x^6}\right)=\frac{s^5}{120},\qquad \mathcal{L}\left((x\cos x-\sin x)^3\right) = \frac{31104-196992 s^2-94080 s^4-13440 s^6}{(s^2+1)^4(s^2+9)^4}$$
turning the original integral into
$$ \frac{1}{120}\int_{0}^{+\infty}\frac{31104s^5-196992 s^7-94080 s^9-13440 s^{11}}{(s^2+1)^4(s^2+9)^4}\,ds $$
that by partial fraction decomposition equals:

$$\color{red}{\frac{4-9\log 3}{40}}$$