show that $\int_{0}^{\infty } \frac{\sin (ax)}{x(x^2+b^2)^2}dx=\frac{\pi}{2b^4}(1-\frac{e^{-ab}(ab+2)}{2})$ show that 
$$\int_{0}^{\infty } \frac{\sin (ax)}{x(x^2+b^2)^2}dx=\frac{\pi}{2b^4}\left(1-\frac{e^{-ab}(ab+2)}{2}\right)$$
for $a,b> 0$
I would like someone solve it using contour but also I would to see different solution using different way to solve it
is there any help 
thanks for all

You can consider employing the residue theorem.  The trick is to use a simple contour despite the pole at $z=0$.  A way to attack this is to deform the usual semicircular contour at the origin so as not to include the pole.
Consider first the case $a>0$.  Rewrite the integral as 
$$\frac12 \int_{-\infty}^{\infty} dx \frac{\sin{a x}}{x (x^2+b^2)^2} = \frac{1}{4 i} \int_{-\infty}^{\infty} dx \frac{e^{i a x} - e^{-i a x}}{x (x^2+b^2)^2}$$
Therefore consider the contour integral
$$\frac{1}{4 i} \oint_C dz \frac{e^{i a z}}{z (z^2+b^2)^2}$$
where $C$ is as described above, a semicircle of radius $R$ in the upper half plane, with a semicircle of radius $\epsilon$ about the origin jutting into the upper half plane.  Note then that the pole at the origin is not within $C$.  

Then the above contour integral is equal to
$$\frac{1}{4 i} \int_{-R}^{-\epsilon} dx \frac{e^{i a x}}{x (x^2+b^2)^2} + \frac{\epsilon}{4} \int_{\pi}^0 d\phi \, e^{i \phi} \frac{e^{i a \epsilon e^{i \phi}}}{\epsilon e^{i \phi} (\epsilon^2 e^{i 2 \phi}+b^2)^2} + \frac{1}{4 i} \int_{\epsilon}^{R} dx \frac{e^{i a x}}{x (x^2+b^2)^2} + \\ \frac{R}{4} \int_0^{\pi} d\theta\, e^{i \theta} \frac{e^{i a R e^{i \theta}}}{R e^{i \theta} (R^2 e^{i 2 \theta}+b^2)^2} $$
We take the limit as $R \to \infty$ and $\epsilon \to 0$ and the contour integral then becomes
$$\frac{1}{4 i} PV \int_{-\infty}^{\infty} dx \frac{e^{i a x}}{x (x^2+b^2)^2} -\frac{\pi}{4 b^4} $$
where $PV$ denotes the Cauchy principal value.  Note that the fourth integral vanishes as 
$$\frac{1}{R^4} \int_0^{\pi} d\theta \, e^{-a R \sin{\theta}} \le \frac{2}{R^4} \int_0^{\pi/2} d\theta \, e^{-2 a R \theta/\pi} = \frac{\pi}{R^5}\frac{1-e^{-a R}}{a} $$
The contour integral is also equal to $(1/(4 i)) i 2 \pi$ times the residue of the integrand at the pole $z=i b$.  Because we have a double pole, the residue has value
$$\begin{align}\frac{d}{dz} \left [ \frac{e^{i a z}}{z (z+i b)^2}\right ]_{z=i b} &= -\left (\frac{a b+2}{4 b^4} \right ) e^{-a b}\end{align}$$
Therefore
$$\frac{1}{4 i} PV \int_{-\infty}^{\infty} dx \frac{e^{i a x}}{x (x^2+b^2)^2} = \frac{\pi}{4 b^4} \left [1-\frac12 (2+a b) e^{-a b} \right ]$$
when $a>0$.  Similarly, one may find that, for $a>0$:
$$\frac{1}{4 i} PV \int_{-\infty}^{\infty} dx \frac{e^{-i a x}}{x (x^2+b^2)^2} = -\frac{\pi}{4 b^4} \left [1-\frac12 (2+a b) e^{-a b} \right ]$$
(Note that the contour integral in this case is $(1/(4 i) (-i 2 \pi)$ times the residue at $z=-i b$ because the contour in the lower half plane is taken clockwise rather than counterclockwise.)  Therefore, for $a>0$,
$$\int_0^{\infty} dx \frac{\sin{a x}}{x (x^2+b^2)^2} = \frac{\pi}{2 b^4} \left [1-\frac12 (2+a b) e^{-a b} \right ]$$
