Optimal. Leaf size=244 \[ -\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (x^{4/3}+x^{2/3}+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^{2/3}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )+1\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )+1\right )-\frac {1}{6} \sin \left (\frac {\pi }{18}\right ) \log \left (x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )+1\right )+\frac {1}{3} \cos \left (\frac {\pi }{18}\right ) \tan ^{-1}\left (\csc \left (\frac {2 \pi }{9}\right ) \left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right )\right )+\frac {1}{3} \sin \left (\frac {\pi }{9}\right ) \tan ^{-1}\left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right )-\frac {1}{3} \sin \left (\frac {2 \pi }{9}\right ) \tan ^{-1}\left (\csc \left (\frac {\pi }{9}\right ) \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )\right ) \]
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Rubi [A] time = 0.30, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {329, 275, 294, 634, 618, 204, 628, 31} \[ -\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (x^{4/3}+x^{2/3}+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^{2/3}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )+1\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )+1\right )-\frac {1}{6} \sin \left (\frac {\pi }{18}\right ) \log \left (x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )+1\right )+\frac {1}{3} \cos \left (\frac {\pi }{18}\right ) \tan ^{-1}\left (\csc \left (\frac {2 \pi }{9}\right ) \left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right )\right )+\frac {1}{3} \sin \left (\frac {\pi }{9}\right ) \tan ^{-1}\left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right )-\frac {1}{3} \sin \left (\frac {2 \pi }{9}\right ) \tan ^{-1}\left (\csc \left (\frac {\pi }{9}\right ) \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 294
Rule 329
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x}}{1-x^6} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^3}{1-x^{18}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{2} \operatorname {Subst}\left (\int \frac {x}{1-x^9} \, dx,x,x^{2/3}\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,x^{2/3}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\frac {1}{2}-\frac {x}{2}}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\cos \left (\frac {\pi }{9}\right )+x \cos \left (\frac {2 \pi }{9}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {-x \cos \left (\frac {\pi }{9}\right )-\sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {-\cos \left (\frac {2 \pi }{9}\right )+x \sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )\\ &=-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^{2/3}\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \operatorname {Subst}\left (\int \frac {2 x-2 \sin \left (\frac {\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \operatorname {Subst}\left (\int \frac {2 x+2 \cos \left (\frac {\pi }{9}\right )}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac {1}{3} \left (\cos \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{6} \sin \left (\frac {\pi }{18}\right ) \operatorname {Subst}\left (\int \frac {2 x-2 \cos \left (\frac {2 \pi }{9}\right )}{1+x^2-2 x \cos \left (\frac {2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac {1}{3} \left (\left (1+\cos \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2-2 x \sin \left (\frac {\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac {1}{3} \left (\sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2+2 x \cos \left (\frac {\pi }{9}\right )} \, dx,x,x^{2/3}\right )\\ &=-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^{2/3}\right )-\frac {1}{3} \left (2 \cos \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {2 \pi }{9}\right )} \, dx,x,2 x^{2/3}-2 \cos \left (\frac {2 \pi }{9}\right )\right )-\frac {1}{3} \left (2 \left (1+\cos \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-x^2-4 \cos ^2\left (\frac {\pi }{18}\right )} \, dx,x,2 x^{2/3}-2 \sin \left (\frac {\pi }{18}\right )\right )+\frac {1}{3} \left (2 \sin \left (\frac {\pi }{9}\right ) \sin \left (\frac {2 \pi }{9}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {\pi }{9}\right )} \, dx,x,2 \left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1+2 x^{2/3}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-x^{2/3}\right )+\frac {1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac {1}{6} \cos \left (\frac {2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac {\pi }{9}\right )\right )+\frac {1}{6} \cos \left (\frac {\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac {\pi }{18}\right )\right )+\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}-\cos \left (\frac {2 \pi }{9}\right )\right ) \csc \left (\frac {2 \pi }{9}\right )\right ) \cot \left (\frac {2 \pi }{9}\right ) \left (1-\sin \left (\frac {\pi }{18}\right )\right )-\frac {1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac {2 \pi }{9}\right )\right ) \sin \left (\frac {\pi }{18}\right )+\frac {1}{3} \tan ^{-1}\left (\sec \left (\frac {\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac {\pi }{18}\right )\right )\right ) \sin \left (\frac {\pi }{9}\right )-\frac {1}{3} \tan ^{-1}\left (\left (x^{2/3}+\cos \left (\frac {\pi }{9}\right )\right ) \csc \left (\frac {\pi }{9}\right )\right ) \sin \left (\frac {2 \pi }{9}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 20, normalized size = 0.08 \[ \frac {3}{4} x^{4/3} \, _2F_1\left (\frac {2}{9},1;\frac {11}{9};x^6\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 555, normalized size = 2.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.59, size = 217, normalized size = 0.89 \[ \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {4}{9} \, \pi \right )}{\sin \left (\frac {4}{9} \, \pi \right )}\right ) \cos \left (\frac {4}{9} \, \pi \right ) \sin \left (\frac {4}{9} \, \pi \right ) + \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} - \cos \left (\frac {2}{9} \, \pi \right )}{\sin \left (\frac {2}{9} \, \pi \right )}\right ) \cos \left (\frac {2}{9} \, \pi \right ) \sin \left (\frac {2}{9} \, \pi \right ) - \frac {2}{3} \, \arctan \left (\frac {x^{\frac {2}{3}} + \cos \left (\frac {1}{9} \, \pi \right )}{\sin \left (\frac {1}{9} \, \pi \right )}\right ) \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \frac {1}{6} \, {\left (\cos \left (\frac {4}{9} \, \pi \right )^{2} - \sin \left (\frac {4}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {4}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {2}{9} \, \pi \right )^{2} - \sin \left (\frac {2}{9} \, \pi \right )^{2}\right )} \log \left (-2 \, x^{\frac {2}{3}} \cos \left (\frac {2}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, {\left (\cos \left (\frac {1}{9} \, \pi \right )^{2} - \sin \left (\frac {1}{9} \, \pi \right )^{2}\right )} \log \left (2 \, x^{\frac {2}{3}} \cos \left (\frac {1}{9} \, \pi \right ) + x^{\frac {4}{3}} + 1\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{\frac {4}{3}} + x^{\frac {2}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | x^{\frac {2}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 162, normalized size = 0.66 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}+1\right )}{6}-\frac {\ln \left (x^{\frac {1}{3}}-1\right )}{6}+\frac {\ln \left (x^{\frac {2}{3}}-x^{\frac {1}{3}}+1\right )}{12}+\frac {\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )}{12}-\frac {\left (-\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )^{3}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )+x^{\frac {1}{3}}\right )}{6 \left (2 \RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )^{5}+\RootOf \left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+1\right )^{2}\right )}+\frac {\left (\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{3}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )+x^{\frac {1}{3}}\right )}{12 \RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{5}-6 \RootOf \left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{\frac {1}{3}} - 1\right )}\right ) + \int \frac {x^{\frac {4}{3}} + 2 \, x^{\frac {1}{3}}}{6 \, {\left (x^{2} + x + 1\right )}}\,{d x} - \int \frac {x^{\frac {4}{3}} - 2 \, x^{\frac {1}{3}}}{6 \, {\left (x^{2} - x + 1\right )}}\,{d x} + \frac {1}{12} \, \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} + 1\right ) + \frac {1}{12} \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left (x^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left (x^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 186, normalized size = 0.76 \[ -\frac {\ln \left (43046721\,x^{2/3}-43046721\right )}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}}{6}+\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,7{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{9}}}{6}-\frac {\ln \left (-43046721\,x^{2/3}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{9}}-43046721\right )\,{\mathrm {e}}^{\frac {\pi \,8{}\mathrm {i}}{9}}}{6}-\ln \left (55788550416\,x^{2/3}\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (55788550416\,x^{2/3}\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^4-43046721\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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