A Beta function related integral

Question

The integral $$\int \limits_{0} ^{1} \left( \frac{x}{x^2+a^2} \right)^p \frac{\log (x)}{x} dx$$ is being considered. Does anyone have an idea how to express it in terms of the Beta function? The differentiation or integration by (any) parameter is allowed.

Answer 1

What immediately comes to mind is to get rid of the log term. To that end, let $t=-\log x$, so that $dx/x=-dt$, $x=e^{-t}$, and $x\in[0,1]\to t\in(\infty,0]$. There follows

$$\int \limits_{0} ^{1} \left( \frac{x}{x^2+a^2} \right)^p \frac{\log (x)}{x} dx=-\int_0^{\infty}\left( \frac{e^{-t}}{e^{-2t}+a^2} \right)^p t dt$$

The following identity is the only one I can find relating the beta functions to exponentials [A. Erdelyi, (1981) Higher Transcendental Functions, Vol. 1 (and particularly, Sec. 1.5), Krieger Publishing.]

$$\int_0^{\infty}e^{-xt}(1-e^{-zt})^{y-1}dt=z^{-1}B(x/z,y),\quad \Re(x/z),\Re(z),\Re(y)>0$$

Unfortunately, the first equation does not fit this format. Perhaps it can be massaged to do so.

On another note, Mathematica gives the following solution for integral

$$\int \limits_{0} ^{1} \left( \frac{x}{x^2+a^2} \right)^p \frac{\log (x)}{x} dx=\frac{(a^2)^{-p}}{p^2}\ _3F_2\left(\frac{p}{2},\frac{p}{2},p;1+\frac{p}{2},1+\frac{p}{2};-\frac{1}{a^2}\right)$$

Perhaps something here will help you get started.