Question Number 138335 by Ñï= last updated on 12/Apr/21
$$\int\frac{{dx}}{{x}^{\mathrm{8}} +{x}^{\mathrm{4}} +\mathrm{1}}=? \\ $$
Answered by MJS_new last updated on 12/Apr/21
$${x}^{\mathrm{8}} +{x}^{\mathrm{4}} +\mathrm{1}= \\ $$$$=\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} −\sqrt{\mathrm{3}}{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +\sqrt{\mathrm{3}}{x}+\mathrm{1}\right) \\ $$$$\mathrm{now}\:\mathrm{simply}\:\mathrm{decompose} \\ $$$$\mathrm{I}\:\mathrm{get} \\ $$$$\frac{\sqrt{\mathrm{3}}}{\mathrm{12}}\left(\mathrm{ln}\:\frac{{x}^{\mathrm{2}} +\sqrt{\mathrm{3}}{x}+\mathrm{1}}{{x}^{\mathrm{2}} −\sqrt{\mathrm{3}}{x}+\mathrm{1}}\:+\mathrm{2}\left(\mathrm{arctan}\:\frac{\mathrm{2}{x}−\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:+\mathrm{arctan}\:\frac{\mathrm{2}{x}+\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)\right)+{C} \\ $$
Commented by Ñï= last updated on 12/Apr/21
$${thank}\:{you}\:{sir}. \\ $$