Optimal. Leaf size=87 \[ \frac {1}{4} \left (1+x^6\right )^{2/3}+\frac {\text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (-1+\sqrt [3]{1+x^6}\right )-\frac {1}{12} \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 0.77, 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 = {272, 52, 57,
632, 210, 31} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{x^6+1}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \left (x^6+1\right )^{2/3}+\frac {1}{4} \log \left (1-\sqrt [3]{x^6+1}\right )-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps
\begin {align*} \int \frac {\left (1+x^6\right )^{2/3}}{x} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {(1+x)^{2/3}}{x} \, dx,x,x^6\right )\\ &=\frac {1}{4} \left (1+x^6\right )^{2/3}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^6\right )\\ &=\frac {1}{4} \left (1+x^6\right )^{2/3}-\frac {\log (x)}{2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^6}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^6}\right )\\ &=\frac {1}{4} \left (1+x^6\right )^{2/3}-\frac {\log (x)}{2}+\frac {1}{4} \log \left (1-\sqrt [3]{1+x^6}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^6}\right )\\ &=\frac {1}{4} \left (1+x^6\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{4} \log \left (1-\sqrt [3]{1+x^6}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 80, normalized size = 0.92 \begin {gather*} \frac {1}{12} \left (3 \left (1+x^6\right )^{2/3}+2 \sqrt {3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^6}\right )-\log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 4.69, size = 64, normalized size = 0.74
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], -x^{6}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )\right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi }\) | \(64\) |
trager | \(\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4}+\frac {\ln \left (\frac {184653895 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-538491663 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+338367746 x^{6}+1600004967 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}+569431707 \left (x^{6}+1\right )^{\frac {2}{3}}-1030573260 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-184653895 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-1600004967 \left (x^{6}+1\right )^{\frac {1}{3}}-754085602 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+845919365}{x^{6}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-111126227 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-300533672 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+98535027 x^{6}+608729953 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}+588476144 \left (x^{6}+1\right )^{\frac {2}{3}}-20253809 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}+111126227 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-608729953 \left (x^{6}+1\right )^{\frac {1}{3}}-477349917 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+131380036}{x^{6}}\right )}{6}\) | \(239\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 63, normalized size = 0.72 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \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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Fricas [A]
time = 0.36, size = 65, normalized size = 0.75 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \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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Sympy [C] Result contains complex when optimal does not.
time = 0.52, size = 36, normalized size = 0.41 \begin {gather*} - \frac {x^{4} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 \Gamma \left (\frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 63, normalized size = 0.72 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \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 = 0.84, size = 89, normalized size = 1.02 \begin {gather*} \frac {\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{4}-\frac {1}{4}\right )}{6}+\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{4}-9\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{4}-9\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {{\left (x^6+1\right )}^{2/3}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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