∫x2−1x4+3x2+1tan−1⁡x+1xdx is equal to

LetI=∫x2−1x4+3x2+1tan−1⁡x+1xdx=∫1−1x2x2+1x2+3tan−1⁡x+1xdxput, x+1x=t⇒1−1x2dx=dtand x+1x2=t2⇒x2+1x2=t2−2∴I=∫1t2+1tan−1⁡tdt=log⁡tan−1⁡t+C    =log⁡tan−1⁡x+1x+C