Can you please help me to solve or write these 2 integrals with some well known functions as gamma, hypergeometric, beta functions,etc...?
$ I= \int_{R}^\infty \frac{(u^{n -1}+1-x^{a/2})}{(1+u^{n -1}x^{a/2})} .dx $
with: n is positive integer, u is positive real number. and 2< a <6.
We can work with R=O, if it's hard to integrate it with every real R.
Many thanks in advance.
PS: the first integral is already solved, $ J= \int_{0}^\infty (\frac{x^{-a/2}}{1+x^{-a/2}})^m .(\frac{1}{1+x^{-a/2}})^n .dx $.
Concerning integral $J$, where we have set $b:=a/2$:
$$J= \displaystyle\int_{0}^\infty \left(\frac{x^{-b}}{1+x^{-b}}\right)^m \left(\frac{1}{1+x^{-b}}\right)^n dx$$
Multiplying numerators and denominators by $x^b$ :
$$J= \displaystyle\int_{x=0}^\infty \left(\frac{1}{1+x^{b}}\right)^m \left(\frac{x^b}{1+x^{b}}\right)^n dx$$
Now, by change of variable $x^b=y \ \iff x=y^{1/b}$:
$$\tag{1}J= \displaystyle\frac{1}{b}\int_{y=0}^\infty \frac{y^{c-1}}{(1+y)^{m+n} } dy$$
with $c$ defined by $c:=n+\frac{1}{b}$.
We recognize in $(1)$ a form of the beta integral ; see formula (10) in (http://homepage.tudelft.nl/11r49/documents/wi4006/gammabeta.pdf):
$$B(u,v)=\displaystyle\int_0^{\infty} \dfrac{s^{u-1}}{(s+1)^{u+v}}ds=\dfrac{\Gamma(u)\Gamma(v)}{\Gamma(u+v)}.$$
giving:
$$J=\frac{1}{b}\dfrac{\Gamma \left(n+\frac{1}{b}\right)\Gamma\left(m-\frac{1}{b}\right)}{\Gamma(m+n)}$$