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:
$$\begin{align*}
I&=\int_{0}^{\infty }{\frac{\sin x}{\cosh x+\cos x}\cdot \frac{{{x}^{n}}}{n!}\ \text{d}x}\tag1 \\ 
 J&= \int_{0}^{\infty }{\frac{x-\sin x}{\left( {{\pi }^{2}}+{{x}^{2}} \right){{x}^{3}}}\ \text{d}x}\tag2 \\ 
\end{align*}$$

Employing the generating function
$$
\frac{\sin x}{\cosh x+\cos x}=\frac12\sum_{k=1}^\infty(-1)^{k-1}e^{-kx}\sin kx
$$
and the classic result of
$$
\int_0^\infty x^{m-1}e^{-ax} \sin bx  \ dx = \frac{\Gamma(m)}{(a^{2} + b^{2})^{m/2}}\sin\left(m\tan^{-1}\left(\frac{b}{a}\right)\right)
$$
The closed-form of the first integral is
\begin{align}
\int_{0}^{\infty}\frac{\sin x}{\cosh x+\cos x}\cdot \frac{x^n}{n!}\ dx&=\frac{2}{n!}\sum_{k=1}^\infty(-1)^{k-1}\int_{0}^{\infty}x^n\ e^{-kx}\sin kx\ dx\\[10pt]
&=\frac{\left(1-2^{-n}\right) \zeta(n+1)}{\ \sqrt{2^{n-1}}}\sin\left(\frac{n+1}{4}\pi\right)
\end{align}
