No partial fractions in the usual sense, but we still managed to break down 1/(x^6+1) and got to the forms that we needed in order to integrate, enjoy!!

Integral of 1/(x^6+1) via partial fraction & Gaussian Elimiation,    • Gaussian Elimination Marathon, #bprpLIVE  
Integral of (x^2+1)/(x^4+1),    • Integral of (x^2+1)/(x^4+1)  
Integral of 1/(x^4+1) from 0 to inf,    • evaluating the integral of 1/(x^4+1) ...  

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