Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx $ using contour integration EDIT: Instead of expressing the integral as the imaginary part of another integral, I instead expanded $\sin^{3}(x)$ in terms of complex exponentials and I don't run into problems anymore.
\begin{align} \int_{0}^{\infty} \frac{x^{3}-\sin^{3}(x)}{x^{5}} \ dx &= \frac{1}{2} \int_{-\infty}^{\infty} \frac{x^{3}-\sin^{3}(x)}{x^{5}} \ dx \\ &= \frac{1}{2} \ \int_{-\infty}^{\infty} \frac{x^{3}+\frac{1}{8i}(e^{3ix}-3e^{ix}+3e^{-ix}-e^{-3ix})}{x^{5}} \ dx \\ &= \frac{1}{2} \lim_{\epsilon \to 0^{+}} \ \int_{-\infty}^{\infty} \frac{x^{3}+\frac{1}{8i}(e^{3ix}-3e^{ix}+3e^{-ix}-e^{-3ix})}{(x-i \epsilon)^{5}} \ dx \\ &= \frac{1}{2} \lim_{\epsilon \to 0^{+}} \int_{-\infty}^{\infty}  \frac{x^{3}+\frac{1}{8i} (e^{3ix}-3e^{ix})}{(x-i \epsilon)^{5}} + \frac{1}{16i} \lim_{\epsilon \to 0^{+}} \int_{-\infty}^{\infty} \frac{3e^{-ix}-e^{-3ix}}{(x-i \epsilon)^{5}} \ dx \end{align}
Then I integrated $ f(z) = \frac{z^{3}+ \frac{1}{8i}(e^{3iz}-3e^{iz})}{(z-i \epsilon)^{5}}$ around the upper half of $|z|=R$ and $ g(z) = \frac{3e^{-iz}-e^{-3iz}}{(z-i \epsilon)^{5}}$ around the lower half of $|z|=R$ and applied Jordan's lemma. 
\begin{align} \int_{0}^{\infty} \frac{x^{3}-\sin^{3}x}{x^{5}} \ dx &= \frac{1}{2} \lim_{\epsilon \to 0^{+}}2 \pi i \ \text{Res}[f(z),i \epsilon] + \frac{1}{16i} \lim_{\epsilon \to 0^{+}} 2 \pi i (0) \\ &= \frac{1}{2} \lim_{\epsilon \to 0^{+}}  \frac{2 \pi i}{4!} \lim_{z \to i \epsilon} \frac{d^{4}}{dz^{4}} \Big(z^{3}+\frac{1}{8i}e^{3iz}-\frac{3}{8i}e^{iz} \Big) \\ &= \frac{\pi i}{24} \lim_{\epsilon \to 0^{+}} \   \lim_{z \to i \epsilon}\Big( \frac{1}{8i}(3i)^{4}e^{3iz}- \frac{3}{8i} (i)^{4} e^{iz} \Big) \\ &= \frac{\pi i}{24} \lim_{\epsilon \to 0^{+}} \Big( \frac{81}{8i}e^{- 3\epsilon} - \frac{3}{8i}e^{- \epsilon} \Big) \\ &= \frac{\pi i}{24} \Big(\frac{81}{8i}-\frac{3}{8i} \Big) \\ &= \frac{13 \pi}{32} \end{align}
 A: Here is another contour integration approach. Note that the integrand is even, so using the contours $\gamma=[-R,R]\cup Re^{i[0,\pi]}$ as $R\to\infty$ and $\beta=[-R,R]\cup Re^{-i[0,\pi]}$, we have
$$
\begin{align}
\int_0^\infty\frac{x^3-\sin^3(x)}{x^5}\mathrm{d}x
&=\frac12\int_{-\infty}^\infty\frac{x^3-\sin^3(x)}{x^5}\mathrm{d}x\tag{1}\\
&=\lim_{\epsilon\to0^+}\frac12\int_{-\infty-i\epsilon}^{\infty-i\epsilon}\frac{z^3-\sin^3(z)}{z^5}\mathrm{d}z\tag{2}\\
&=\lim_{\epsilon\to0^+}\frac12\int_{\gamma-i\epsilon}\frac{\frac1{8i}(e^{3iz}-3e^{iz})}{z^5}\mathrm{d}z\tag{3}\\
&+\lim_{\epsilon\to0^+}\frac12\int_{\beta-i\epsilon}\frac{z^3+\frac1{8i}(3e^{-iz}-e^{-3iz})}{z^5}\mathrm{d}z\tag{4}\\
&=\lim_{\epsilon\to0^+}\frac1{16i}\int_{\gamma-i\epsilon}\frac{e^{3iz}-3e^{iz}}{z^5}\mathrm{d}z\tag{5}\\[3pt]
&=\frac\pi8\frac1{4!}\left((3i)^4-3i^4\right)\tag{6}\\[6pt]
&=\frac{13\pi}{32}\tag{7}
\end{align}
$$
Explanation:
$(1)$: integrand is even, duplicate the domain and divide by $2$
$(2)$: offset the path of integration since there are no singularities and the integrand decays at $\infty$
$(3)$: take some of the terms along the upper contour
$(4)$: take the rest along the lower contour
$(5)$: there are no singularities inside $\beta-i\epsilon$ and the respective integrands decay appropriately on the circular arcs
$(6)$: the residues depend on the $z^4$ terms in the expansions of the exponentials
A: Another approach :
Sorry Random Variable, this is not using contour integration technique since I don't know how to approach the integral using that way. $\ddot\smile$
Consider
$$
\mathcal{I}(\alpha)=\int_0^\infty\frac{(\alpha x)^3-\sin^3\alpha x}{x^5}dx.\tag1
$$
Differentiating $(1)$ four times yields
\begin{align}
\frac{d^4\mathcal{I}}{d\alpha^4}&=\int_0^\infty\frac{\partial^4}{\partial\alpha^4}\left(\frac{(\alpha x)^3-\sin^3\alpha x}{x^5}\right)dx\\
&=\color{green}{\int_0^\infty\left(\frac{81\sin3\alpha x-3\sin\alpha x}{4x}\right)dx}\\
&=\frac{81}{4}\cdot\frac\pi2-\frac{3}{4}\cdot\frac\pi2\\
&=\frac{39\pi}{4},\tag2
\end{align}
where
$$
\int_0^\infty\frac{\sin\alpha x}{x}dx=\frac\pi2\qquad\text{for }\alpha\neq0.
$$
Then from $(2)$ we obtain
$$
\large\color{blue}{\mathcal{I}(\alpha)=\frac{13\pi}{32}a^4},\tag3
$$
where $\mathcal{I}(0)=\mathcal{I'}(0)=\mathcal{I''}(0)=\mathcal{I'''}(0)=0$. Thus

$$
\mathcal{I}(1)=\int_0^\infty\frac{x^3-\sin^3 x}{x^5}dx=\large\color{blue}{\frac{13\pi}{32}}.
$$


P.S.
I think you can easily apply the contour integration technique in line $2$ (green-colored) equation $(2)$.
A: I like to calculate this integral as follows:  
Let us note that  
$$\frac{1}{x^5}=\frac{1}{4!}\int_0^\infty t^4e^{-xt}dt$$ So  
$$I=\frac{1}{4!}\int_{0}^{\infty}(x^{3}-\sin^{3}x)\int_0^\infty t^4e^{-xt}\;dt\;dx$$  
$$=\frac{1}{4!}\int_{0}^{\infty}t^4\int_{0}^{\infty}(x^{3}-\sin^{3}x)e^{-xt}\;dx\;dt$$   
$$=\frac{1}{4!} \int_{0}^{\infty}t^4\left [\frac{6}{t^4}-\frac{6}{(t^2+1)(t^2+9)}\right ]dt$$   
$$=\frac{1}{4}\int_{0}^{\infty}\frac{10t^2+9}{(t^2+1)(t^2+9)}dt=\frac{13\pi}{32}$$
A: $\newcommand{\+}{^{\dagger}}
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$\ds{\int_{0}^{\infty}{x^{3}- \sin^{3}\pars{x} \over x^{5}}\,\dd x:\ {\large ?}}$

Lets
  $\ds{\color{#00f}{\fermi\pars{x}} \equiv x^{3} - \sin^{3}\pars{x}
     = \color{#00f}{x^{3} + {1 \over 4}\,\sin\pars{3x}
       - {3 \over 4}\,\sin\pars{x}
       }\tag{1}}$.

$$
\mbox{The integral in question becomes}\quad
\int_{0}^{\infty}{\fermi\pars{x} \over x^{5}}\,\dd x
$$

In order to 'reduce' the $\ds{x^{-5}}$ power to a 'simple' $\ds{x^{-1}}$ power, we integrate by parts repeatedly:
  \begin{align}
\color{#c00000}{\int_{0}^{\infty}{\fermi\pars{x} \over x^{5}}\,\dd x}&={1 \over 4}\int_{0}^{\infty}{\fermi'\pars{x} \over x^{4}}\,\dd x
={1 \over 12}\int_{0}^{\infty}{\fermi''\pars{x} \over x^{3}}\,\dd x={1 \over 24}\int_{0}^{\infty}{\fermi'''\pars{x} \over x^{2}}\,\dd x
\\[3mm]&={1 \over 24}\int_{0}^{\infty}{\fermi^{\pars{\tt IV}}\pars{x} \over x}
\,\dd x\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\pars{2}
\end{align}

From expression $\pars{1}$ we can evaluate $\ds{\fermi^{\pars{\tt IV}}\pars{x}}$:
$$
\fermi^{\pars{\tt IV}}\pars{x}
={81 \over 4}\,\sin\pars{3x} - {3 \over 4}\,\sin\pars{x}
$$

which is replaced in $\pars{2}$:
  \begin{align}
\color{#c00000}{\int_{0}^{\infty}{\fermi\pars{x} \over x^{5}}\,\dd x}&
={27 \over 32}\int_{0}^{\infty}{\sin\pars{3x} \over x}\,\dd x
-{1 \over 32}\int_{0}^{\infty}{\sin\pars{x} \over x}\,\dd x
\\[3mm]&={27 \over 32}\int_{0}^{\infty}{\sin\pars{x} \over x}\,\dd x
-{1 \over 32}\int_{0}^{\infty}{\sin\pars{x} \over x}\,\dd x
\\[3mm]&=\underbrace{\pars{{27 \over 32} - {1 \over 32}}}_{\ds{13 \over 16}}\
\underbrace{\int_{0}^{\infty}{\sin\pars{x} \over x}\,\dd x}_{\ds{\pi \over 2}}
\end{align}

$$\color{#00f}{\large%
\int_{0}^{\infty}{x^{3}- \sin^{3}\pars{x} \over x^{5}}\,\dd x
={13 \over 32}\,\pi} \approx {\tt 1.2763}
$$
