$\int \frac{2x^2-1+x^2\sqrt{x^2+4}}{x^2\left(x^2+4\right)}dx$

$\frac{9}{8}{\tan }^{-1}\frac{x}{2}+\frac{1}{4x}+{\cosh }^{-1}\frac{x}{2}+c$

$\frac{9}{8}{\tan }^{-1}\frac{x}{2}+\frac{1}{4x}+{\sinh }^{-1}\frac{x}{2}+c$

$\frac{9}{16}\log \left|\frac{x+2}{x-2}\right|+\frac{1}{4x}+\log \left|\frac{x+\sqrt{x^2+4}}{2}\right|+c$

$\frac{9}{16}\log \left|\frac{2-x}{2+x}\right|+\frac{1}{4x}+{\cosh }^{-1}\frac{x}{2}+c$