Efficient method to evaluate integrals with multiple roots in denominator $$ \int \frac{x^2-1}{x\sqrt{(x^2 + \beta x +1 )} \sqrt{(x^2 + \alpha x +1 )}} dx $$
I saw this question but I don't think the methods in answers are much applicable here.
My attempt:
$$ \int \frac{x^2-1}{x\sqrt{(x + \frac{\beta}{2})^2  + ( 1 - \frac{\beta^2}{4} )} \sqrt{(x + \frac{\alpha}{2})^2  + ( 1 - \frac{\alpha^2}{4} )}} dx= \frac{1}{\sqrt{ 1- \frac{\beta^2}{4} } \sqrt{1- \frac{\alpha^2}{4} }}\int \frac{x^2-1}{x\sqrt{\frac{(x + \frac{\beta}{2})^2}{ ( 1 - \frac{\beta^2}{4} )}  +1} \sqrt{\frac{(x + \frac{\alpha}{2})^2}{ ( 1 - \frac{\alpha^2}{4} )}  +1}} dx
$$
Not sure what's the best way to go now..

Divide both numerator and denominator by $x^2$
$$\int \frac{1-\frac{1}{x^2}}{\sqrt{(x+\beta +\frac{1}{x})}\sqrt{(x+\alpha +\frac{1}{x})}}dx$$
which immediately suggests using the substitution $t=x+\frac{1}{x}$
$$\int \frac{dt}{\sqrt{\left(t+\frac{\alpha+\beta}{2}\right)^2-\left(\frac{\alpha-\beta}{2}\right)^2}} = \cosh^{-1}\left[\frac{t+\frac{\alpha+\beta}{2}}{\left|\frac{\alpha-\beta}{2}\right|}\right]$$
from a simple hyperbolic substitution. This makes the final answer
$$\cosh^{-1}\left[\frac{x+\frac{1}{x}+\frac{\alpha+\beta}{2}}{\left|\frac{\alpha-\beta}{2}\right|}\right]$$
