How to integrate $\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x} $? I have done one with $\displaystyle\int_0^{\infty}\frac{x-\sin x}{x^3}\ \text{d}x$, but I have no ideas with these:
$$\begin{align*}
I&=\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x}\tag1 \\ 
 J&= \int_{0}^{\infty }{\frac{x-\sin x}{\left( {{\pi }^{2}}+{{x}^{2}} \right){{x}^{3}}}\ \text{d}x}\tag2 \\ 
\end{align*}$$

Maple says
$$
\int_{0}^{\infty} \frac{x - \operatorname{sin} (x)}{\bigl(\pi^{2} + x^{2}\bigr) x^{3}} d x = \frac{2 \operatorname{sinh} (\pi) - 2 \pi - 2 \operatorname{cosh} (\pi) + \pi^{2} + 2}{4 \pi^{3}}
\approx 0.04434578936
$$
but it didn't do the first one automatically
