We have,
I=∫x2x4+x2+1dx
I=∫x2(x2−x+1)(x2+x+1)dx
I=12∫x(x2−x+1)dx−12∫x(x2+x+1)dx
I=14∫2x−1(x2−x+1)dx+14∫1(x2−x+1)dx−14∫2x+1(x2+x+1)dx+14∫1(x2+x+1)dx
I=14∫2x−1(x2−x+1)dx+14∫1(x−12)2+34dx−14∫2x+1(x2+x+1)dx+14∫1(x+12)2+34dx
I=14∫2x−1(x2−x+1)dx+14∫1(x−12)2+(√32)2dx−14∫2x+1(x2+x+1)dx+14∫1(x+12)2+(√32)2dx
I=14ln(x2−x+1)+14⎡⎢ ⎢ ⎢ ⎢⎣1√32tan−1⎛⎜ ⎜ ⎜ ⎜⎝x−12√32⎞⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥⎦−14ln(x2+x+1)+14⎡⎢ ⎢ ⎢ ⎢⎣1√32tan−1⎛⎜ ⎜ ⎜ ⎜⎝x+12√32⎞⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥⎦+C
I=14ln(x2−x+1)+12√3tan−1(2x−1√3)−14ln(x2+x+1)+12√3tan−1(2x+1√3)+C
I=14ln(x2−x+1)−14ln(x2+x+1)+12√3tan−1(2x−1√3)+12√3tan−1(2x+1√3)+C
Hence, this is the answer.