Question Number 120599 by MagdiRagheb last updated on 01/Nov/20
$${Let}\:{x}\:=\:{u}^{\mathrm{6}} \:\:\:\:\:\:\:\:\:{dx}\:=\:\mathrm{6}{u}^{\mathrm{5}} \:{du} \\ $$$${I}\:=\:\int\frac{{u}^{\mathrm{3}} }{\left(\mathrm{1}+{u}^{\mathrm{2}} \right)^{\mathrm{2}} }\:×\mathrm{6}{u}^{\mathrm{5}} \:{du} \\ $$$$\:\:\:=\mathrm{6}\:\int\frac{{u}^{\mathrm{8}} }{\mathrm{1}+\mathrm{2}{u}^{\mathrm{2}} +{u}^{\mathrm{4}} }\:{du} \\ $$$$\:\:\:=\mathrm{6}\:\int\left[\frac{−\mathrm{4}}{{u}^{\mathrm{2}} +\mathrm{1}}+\frac{\mathrm{1}}{\left(\mathrm{1}+{u}^{\mathrm{2}} \right)^{\mathrm{2}} }+{u}^{\mathrm{4}} −\mathrm{2}{u}^{\mathrm{2}} +\mathrm{3}\right]{du} \\ $$$$\:\:\:=\mathrm{6}\left[−\mathrm{4}{tan}^{−\mathrm{1}} \left({u}\right)\:+\:{I}_{\mathrm{1}} \:+\:\frac{{u}^{\mathrm{5}} }{\mathrm{5}}\:−\:\frac{\mathrm{2}}{\mathrm{3}}{u}^{\mathrm{3}} \:+\mathrm{3}{u}\right]+{c} \\ $$$${I}_{\mathrm{1}} \:=\:\int\frac{\mathrm{1}}{\left(\mathrm{1}+{u}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{du} \\ $$$${Put}\:{u}\:=\:{tan}\:{z}\:\:\:\:\:{du}\:=\:{sec}^{\mathrm{2}} {z}\:{dz} \\ $$$${I}_{\mathrm{1}} \:=\:\int\frac{\mathrm{1}}{{sec}^{\mathrm{4}} {z}}×{sec}^{\mathrm{2}} {z}\:{dz}\:=\:\int{cos}^{\mathrm{2}} {z}\:{dz} \\ $$$$\:\:\:\:\:=\:\int\frac{\mathrm{1}}{\mathrm{2}}\left({cos}\:\mathrm{2}{z}\:+\:\mathrm{1}\right){dz} \\ $$$$\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}\:{sin}\:\mathrm{2}{z}\:+\:{z}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}×\frac{{u}}{\mathrm{1}+{u}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} {u} \\ $$$${I}\:=\:−\mathrm{4}\:{tan}^{−\mathrm{1}} {u}\:+\:\frac{{u}}{\mathrm{2}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} {u}\:+\frac{{u}^{\mathrm{5}} }{\mathrm{5}}\:−\:\frac{\mathrm{2}}{\mathrm{3}}{u}^{\mathrm{3}} \:+\:\mathrm{3}{u}\:+\:{c} \\ $$$$\:\:\:=\:−\frac{\mathrm{7}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} {u}\:+\:\frac{{u}}{\mathrm{2}\left(\mathrm{1}+{u}^{\mathrm{2}} \right)}\:+\:\frac{\mathrm{1}}{\mathrm{5}}{u}^{\mathrm{5}} \:+\:\mathrm{3}{u}\:+\:{c} \\ $$$$\:\:\:=\:−\frac{\mathrm{7}}{\mathrm{2}}\:{tan}^{−\mathrm{1}} \left(\overset{\mathrm{6}} {\:}\sqrt{{x}}\right)\:+\:\frac{\sqrt[{\mathrm{6}}]{{x}}}{\mathrm{2}\left(\mathrm{1}+\sqrt[{\mathrm{6}}]{{x}^{\mathrm{2}} }\right)}\:+\:\frac{\mathrm{1}}{\mathrm{5}}\:\sqrt[{\mathrm{6}}]{{x}^{\mathrm{5}} }\:+\:\mathrm{3}\:\sqrt[{\mathrm{6}}]{{x}}\:+\:{c} \\ $$