Definite logarithmic integral

Let $0<\alpha<\beta$ . Evaluate the integral:

$\mathcal{J}=\int_\alpha^\beta \frac{\ln x}{(x+\alpha)(x+\beta)}\, \mathrm{d}x$

Solution

We have successively:

$\begin{align*} \mathcal{J} &= \int_{\alpha}^{\beta} \frac{\ln x}{\left ( x+\alpha \right )\left ( x+\beta \right )}\, \mathrm{d}x \\ &\!\!\!\!\!\!\!\overset{x \mapsto \alpha \beta/x}{=\! =\! =\! =\! =\!=\!} \int_{\alpha}^{\beta} \frac{\ln \alpha \beta- \ln x}{\left ( x + \alpha \right ) \left ( x + \beta \right )} \, \mathrm{d}x\\ &=\frac{1}{2} \int_{\alpha}^{\beta} \frac{\ln \alpha \beta}{\left ( x+\alpha \right )\left ( x + \beta \right )} \, \mathrm{d}x \\ &=\frac{\ln \alpha \beta}{\beta-\alpha} \ln \left ( \frac{(\alpha+\beta)^2}{4\alpha \beta} \right ) \end{align*}$