integration

this one is easy
\int \frac{x^{4}-1}{x^{2}x\left(x^{2}+1+\frac{1}{x^{2}} \right)^{1/2}}dx
\Rightarrow \int \frac{x-\frac{1}{x^{3}}}{\left( x^{2}+1+\frac{1}{x^{2}}\right)^{1/2}}dx

\Rightarrow \frac{1}2{}\int \frac{2x-\frac{2}{x^{3}}}{\left( x^{2}+1+\frac{1}{x^{2}}\right)^{1/2}}dx

Put \left( x^{2}+1+\frac{1}{x^{2}}\right)= t^{2}\Rightarrow 2x-\frac{2}{x^{3}}=2tdt

now,
\frac{1}{2 }\int 2dt = t + c
\Rightarrow \sqrt{\left(x^{2}+1+\frac{1}{x^{2}} \right)}+c