∫1/(x⁴+x²+1) dx
= (1/2)∫(x+1)/(x²+x+1) dx - (1/2)∫(x-1)/(x²-x+1) dx
= (1/2)[(1/2)∫(2x+1)/(x²+x+1) dx + (1/2)∫1/(x²+x+1) dx]
- (1/2)[(1/2)∫(2x-1)/(x²-x+1) dx - (1/2)∫1/(x²-x+1) dx]
= (1/4)∫d(x²+x+1)/(x²+x+1) - (1/4)∫d(x²-x+1)/(x²-x+1)
+ (1/4)∫d(x+1/2)/[(x+1/2)²+3/4] + (1/4)∫d(x-1/2)/[(x-1/2)²+3/4]
= (1/4)ln|(x²+x+1)/(x²-x+1)| + (2/√3)arctan[(2x+1)/√3] + (2/√3)arctan[(2x-1)√3] + C