Optimal. Leaf size=90 \[ -\frac {\left (x^6+1\right )^{2/3}}{6 x^6}+\frac {1}{9} \log \left (\sqrt [3]{x^6+1}-1\right )-\frac {1}{18} \log \left (\left (x^6+1\right )^{2/3}+\sqrt [3]{x^6+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^6+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 47, 55, 618, 204, 31} \begin {gather*} -\frac {\left (x^6+1\right )^{2/3}}{6 x^6}+\frac {1}{6} \log \left (1-\sqrt [3]{x^6+1}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^6+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 55
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (1+x^6\right )^{2/3}}{x^7} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{6 x^6}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^6\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{6 x^6}-\frac {\log (x)}{3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^6}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^6}\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{6 x^6}-\frac {\log (x)}{3}+\frac {1}{6} \log \left (1-\sqrt [3]{1+x^6}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^6}\right )\\ &=-\frac {\left (1+x^6\right )^{2/3}}{6 x^6}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{6} \log \left (1-\sqrt [3]{1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.29 \begin {gather*} \frac {1}{10} \left (x^6+1\right )^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};x^6+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 90, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^6\right )^{2/3}}{6 x^6}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-1+\sqrt [3]{1+x^6}\right )-\frac {1}{18} \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 79, normalized size = 0.88 \begin {gather*} \frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{18 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 66, normalized size = 0.73 \begin {gather*} \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} - \frac {1}{18} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.64, size = 76, normalized size = 0.84
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+6 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{18 \pi }\) | \(76\) |
risch | \(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{6 x^{6}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi }\) | \(76\) |
trager | \(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{6 x^{6}}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-143971236 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+255097463 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+65690018 x^{6}+608729953 \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+20253809 \left (x^{6}+1\right )^{\frac {2}{3}}+608729953 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}+143971236 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+20253809 \left (x^{6}+1\right )^{\frac {1}{3}}+752701189 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+164225045}{x^{6}}\right )}{9}-\frac {\ln \left (-\frac {353837768 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+1230697177 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+507551619 x^{6}+1600004967 \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1030573260 \left (x^{6}+1\right )^{\frac {2}{3}}+1600004967 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-353837768 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+1030573260 \left (x^{6}+1\right )^{\frac {1}{3}}+1246167199 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+676735492}{x^{6}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{9}-\frac {\ln \left (-\frac {353837768 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+1230697177 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+507551619 x^{6}+1600004967 \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1030573260 \left (x^{6}+1\right )^{\frac {2}{3}}+1600004967 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-353837768 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+1030573260 \left (x^{6}+1\right )^{\frac {1}{3}}+1246167199 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+676735492}{x^{6}}\right )}{9}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 66, normalized size = 0.73 \begin {gather*} \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} - \frac {1}{18} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 92, normalized size = 1.02 \begin {gather*} \frac {\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{9}-9\,{\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{9}-9\,{\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^6+1\right )}^{2/3}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.09, size = 32, normalized size = 0.36 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 x^{2} \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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