The integral $\underset{2}{\overset{3}{\int }}\frac{2x^5+x^4-2x^3+2x^2+1}{(x^2+1)(x^4-1)}dx$ is equal to 

Nr = $2x^3(x^2-1)+{(x^2+1)}^2$

$(\sec x+Tanx)\Rightarrow I=\underset{2}{\overset{3}{\int }}[\frac{2x^3}{{(x^2+1)}^2}+\frac{1}{x^2-1}]dx=\underset{2}{\overset{3}{\int }}\frac{x^2}{{(x^2+1)}^2}2xdx+\underset{2}{\overset{3}{\int }}\frac{1}{(x^2-1)}$

$\Rightarrow I_1=\underset{5}{\overset{10}{\int }}\frac{t-1}{t^2}\text{\hspace{0.17em}}dt={\{\log t+\frac{1}{t}\}}_5^{10} \& I_2=\frac{1}{2}\log \left(\frac{3}{2}\right)$

$=\frac{1}{2}\log 6-\frac{1}{10}$