Exact Values of the integal $\int_0^\infty \frac{r^{n-1}}{(1+r^2)^{\frac{s}{2}}}\,dr$ Does any one know the exact expression of the integral,
$$E_n(s)=\int_0^\infty \frac{r^{n-1}}{(1+r^2)^{\frac{s}{2}}}\,dr~~~~s>n, n\in \mathbb{N}$$
or more generally,
$$E_a(s)=\int_0^\infty \frac{r^{a-1}}{(1+r^2)^{\frac{s}{2}}}\,dr~~~~s>a, a\in \mathbb{R}$$
For the special case $s=n, n+2$ I find out by induction that
$$ E_{n-1}(n)=\frac{\omega_{n-1}}{2\omega_{n-2}}~~\text{and}~~E_{n-1}(n+2)=\frac{\omega_{n-1}}{2n\omega_{n-2}}.
$$
where $\omega_{n-1} = \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}$ is the  surface measure of the n-dimensional sphere of $\mathbb{R}^n$.
Further result is welcome

By setting $\frac{1}{1+r^2}=u$ we get that $E_n(s)$ depends on a value of the Beta function:
$$ E_n(s) = \frac{\Gamma\left(\frac{n}{2}\right)\,\Gamma\left(\frac{s-n}{2}\right)}{2\,\Gamma\left(\frac{s}{2}\right)}.\tag{1} $$
