I would like to study how to evaluate the definite integral$$\int_{-\infty}^\infty \frac{x-2}{(x^2+x+4)\sqrt{x^2+2x+4}}dx$$
Based on the integration techniques that I know, I proceed by completing the square and then introducing the trigonometric substitution $x+1=\sqrt3 \sinh t$ to transform the integral to$$ \int_{-\infty}^\infty \frac{\sqrt3 \sinh t-3 }{3\sinh^2t-\sqrt3\sinh t +4}dt$$At this point, I further introduce the ‘half-angle’ substitution $y=\tanh\frac t2$. But, the resulting integrand is of a quartic function in $y$, which I do not how how to deal with. I prefer using real method.
