Optimal. Leaf size=74 \[ \frac {\text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (-1+\sqrt [3]{1+x^6}\right )-\frac {1}{12} \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 0.73, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {272, 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} \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 57
Rule 210
Rule 272
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{1+x^6}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^6\right )\\ &=-\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 {\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 {\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 = 69, normalized size = 0.93 \begin {gather*} \frac {1}{12} \left (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 = 4.78, size = 63, normalized size = 0.85
method | result | size |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{12 \pi }\) | \(63\) |
trager | \(\frac {\ln \left (\frac {184653895 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-538491663 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+338367746 x^{6}+1600004967 \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+569431707 \left (x^{6}+1\right )^{\frac {2}{3}}-1030573260 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}-184653895 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-1600004967 \left (x^{6}+1\right )^{\frac {1}{3}}-754085602 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+845919365}{x^{6}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-169183873 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-322897724 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+553961685 x^{6}+1600004967 \left (x^{6}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1030573260 \left (x^{6}+1\right )^{\frac {2}{3}}-569431707 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}+169183873 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-1600004967 \left (x^{6}+1\right )^{\frac {1}{3}}-861389387 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+738615580}{x^{6}}\right )}{6}\) | \(230\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 54, normalized size = 0.73 \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}{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 = 0.79, size = 56, normalized size = 0.76 \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}{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 = 0.41, size = 31, normalized size = 0.42 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{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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Giac [A]
time = 0.42, size = 54, normalized size = 0.73 \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}{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 = 0.86, size = 80, normalized size = 1.08 \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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