$\large \int \frac{x dx}{\sqrt{x^{4}-1}}$
lets take $\large z=x^{2}$
$\large \frac{\mathrm{d}z }{\mathrm{d} x}=2x$
$\large \frac{dz}{2}=xdx$
substituting it in the integral,
$\large \int \frac{dz}{2\sqrt{z^{2}-1}}$
=$\large \frac{1}{2}\ln\left | z+\sqrt{z^{2}-1} \right |+c$ as, $\large \left [ \int \frac{dx}{\sqrt{x^{2}-a^{2}}} \right ]=\ln \left | x+\sqrt{x^{2}-a^{2}} \right |+c$
=$\large \frac{1}{2}\ln\left | x^{2}+\sqrt{x^{4}-1} \right |+c$