Optimal. Leaf size=112 \[ \frac {1}{36} \left (1+x^6\right )^{2/3} \left (-4 x^2+3 x^8\right )+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1+x^6}}\right )}{9 \sqrt {3}}-\frac {1}{27} \log \left (-x^2+\sqrt [3]{1+x^6}\right )+\frac {1}{54} \log \left (x^4+x^2 \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 0.76, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 245}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{12} \left (x^6+1\right )^{2/3} x^8-\frac {1}{9} \left (x^6+1\right )^{2/3} x^2-\frac {1}{18} \log \left (x^2-\sqrt [3]{x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 281
Rule 327
Rubi steps
\begin {align*} \int \frac {x^{13}}{\sqrt [3]{1+x^6}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^6}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}-\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=-\frac {1}{9} x^2 \left (1+x^6\right )^{2/3}+\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=-\frac {1}{9} x^2 \left (1+x^6\right )^{2/3}+\frac {1}{12} x^8 \left (1+x^6\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{1+x^6}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{18} \log \left (x^2-\sqrt [3]{1+x^6}\right )\\ \end {align*}
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Mathematica [A]
time = 1.70, size = 107, normalized size = 0.96 \begin {gather*} \frac {1}{108} \left (3 x^2 \left (1+x^6\right )^{2/3} \left (-4+3 x^6\right )+4 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{1+x^6}}\right )-4 \log \left (-x^2+\sqrt [3]{1+x^6}\right )+2 \log \left (x^4+x^2 \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 = 3.53, size = 17, normalized size = 0.15
method | result | size |
meijerg | \(\frac {x^{14} \hypergeom \left (\left [\frac {1}{3}, \frac {7}{3}\right ], \left [\frac {10}{3}\right ], -x^{6}\right )}{14}\) | \(17\) |
risch | \(\frac {x^{2} \left (3 x^{6}-4\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{36}+\frac {x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{6}\right )}{9}\) | \(37\) |
trager | \(\frac {x^{2} \left (3 x^{6}-4\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{36}-\frac {\ln \left (-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-2 x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}+1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{27}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{4}-2 x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right )}{27}\) | \(228\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 122, normalized size = 1.09 \begin {gather*} -\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {7 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} - \frac {4 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}}}{x^{10}}}{36 \, {\left (\frac {2 \, {\left (x^{6} + 1\right )}}{x^{6}} - \frac {{\left (x^{6} + 1\right )}^{2}}{x^{12}} - 1\right )}} + \frac {1}{54} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{27} \, \log \left (\frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 102, normalized size = 0.91 \begin {gather*} \frac {1}{36} \, {\left (3 \, x^{8} - 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} - \frac {1}{27} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {1}{27} \, \log \left (-\frac {x^{2} - {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{54} \, \log \left (\frac {x^{4} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}}\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 = 1.64, size = 29, normalized size = 0.26 \begin {gather*} \frac {x^{14} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {10}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{13}}{{\left (x^6+1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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