Optimal. Leaf size=117 \[ \frac {7}{243} \log \left (\sqrt [3]{x^6+1}-x^2\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6+1}+x^2}\right )}{81 \sqrt {3}}-\frac {7}{486} \log \left (\left (x^6+1\right )^{2/3}+x^4+\sqrt [3]{x^6+1} x^2\right )+\frac {1}{324} \left (x^6+1\right )^{2/3} \left (18 x^{14}-21 x^8+28 x^2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 101, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 321, 239} \begin {gather*} \frac {1}{18} \left (x^6+1\right )^{2/3} x^{14}-\frac {7}{108} \left (x^6+1\right )^{2/3} x^8+\frac {7}{81} \left (x^6+1\right )^{2/3} x^2+\frac {7}{162} \log \left (x^2-\sqrt [3]{x^6+1}\right )-\frac {7 \tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^{19}}{\sqrt [3]{1+x^6}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{18} x^{14} \left (1+x^6\right )^{2/3}-\frac {7}{18} \operatorname {Subst}\left (\int \frac {x^6}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=-\frac {7}{108} x^8 \left (1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (1+x^6\right )^{2/3}+\frac {7}{27} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {7}{81} x^2 \left (1+x^6\right )^{2/3}-\frac {7}{108} x^8 \left (1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (1+x^6\right )^{2/3}-\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^2\right )\\ &=\frac {7}{81} x^2 \left (1+x^6\right )^{2/3}-\frac {7}{108} x^8 \left (1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (1+x^6\right )^{2/3}-\frac {7 \tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{1+x^6}}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {7}{162} \log \left (x^2-\sqrt [3]{1+x^6}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 127, normalized size = 1.09 \begin {gather*} \frac {1}{972} \left (54 \left (x^6+1\right )^{2/3} x^{14}-63 \left (x^6+1\right )^{2/3} x^8+84 \left (x^6+1\right )^{2/3} x^2+28 \log \left (1-\frac {x^2}{\sqrt [3]{x^6+1}}\right )-28 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6+1}}+1}{\sqrt {3}}\right )-14 \log \left (\frac {x^4}{\left (x^6+1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6+1}}+1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 7.52, size = 117, normalized size = 1.00 \begin {gather*} \frac {7}{243} \log \left (\sqrt [3]{x^6+1}-x^2\right )-\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6+1}+x^2}\right )}{81 \sqrt {3}}-\frac {7}{486} \log \left (\left (x^6+1\right )^{2/3}+x^4+\sqrt [3]{x^6+1} x^2\right )+\frac {1}{324} \left (x^6+1\right )^{2/3} \left (18 x^{14}-21 x^8+28 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 107, normalized size = 0.91 \begin {gather*} \frac {1}{324} \, {\left (18 \, x^{14} - 21 \, x^{8} + 28 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \frac {7}{243} \, \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 {7}{243} \, \log \left (-\frac {x^{2} - {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {7}{486} \, \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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{19}}{{\left (x^{6} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 42, normalized size = 0.36 \begin {gather*} \frac {x^{2} \left (18 x^{12}-21 x^{6}+28\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{324}-\frac {7 x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{6}\right )}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 145, normalized size = 1.24 \begin {gather*} \frac {7}{243} \, \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 {67 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{x^{4}} - \frac {77 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}}}{x^{10}} + \frac {28 \, {\left (x^{6} + 1\right )}^{\frac {8}{3}}}{x^{16}}}{324 \, {\left (\frac {3 \, {\left (x^{6} + 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} + 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} + 1\right )}^{3}}{x^{18}} - 1\right )}} - \frac {7}{486} \, \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 {7}{243} \, \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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{19}}{{\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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sympy [C] time = 2.30, size = 29, normalized size = 0.25 \begin {gather*} \frac {x^{20} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {13}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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