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NotSObright
integral((x^3+x+1)/(x^4+x^2+1))
http://www.wolframalpha.com/input/?i=integral%28%28x^3%2Bx%2B1%29%2F%28x^4%2Bx^2%2B1%29%29
I am concerned with what substitution has to be carried out and not the answer
you have the form P(x)/Q(x) where P and Q are polynomials. Further the degree of the numerator, namely 3, is one less than the denominator. In such situations, find \[dQ/dx \] and express P in terms of that. \[\int\limits{dy/y} = \log y\] the remaining terms are hopefully simpler. If you need more help ask
\[d/dx ({{x^4+x^2+1}}) = 4x^3 + 2x \] so \[P(x) = {1/4}({dQ/dx}) + (1/2)x + 1 \] So the original problem reduces to integrating \[dQ/Q + (1/2)\int\limits x dx/Q + \int\limits dx/Q \]
Sorry the last equation should read: \[(1/4)\int\limits dQ/Q + (1/2)\int\limits\limits x dx/Q + \int\limits\limits dx/Q\]