Optimal. Leaf size=117 \[ \frac {5}{486} \log \left (\sqrt [3]{x^6-1}-x^2\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6-1}+x^2}\right )}{162 \sqrt {3}}-\frac {5}{972} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right )+\frac {1}{324} \sqrt [3]{x^6-1} \left (18 x^{16}-3 x^{10}-5 x^4\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 135, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {275, 279, 321, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{18} \sqrt [3]{x^6-1} x^{16}-\frac {1}{108} \sqrt [3]{x^6-1} x^{10}-\frac {5}{324} \sqrt [3]{x^6-1} x^4+\frac {5}{486} \log \left (1-\frac {x^2}{\sqrt [3]{x^6-1}}\right )+\frac {5 \tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{162 \sqrt {3}}-\frac {5}{972} \log \left (\frac {x^4}{\left (x^6-1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6-1}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 279
Rule 292
Rule 321
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int x^{15} \sqrt [3]{-1+x^6} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^7 \sqrt [3]{-1+x^3} \, dx,x,x^2\right )\\ &=\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}-\frac {1}{18} \operatorname {Subst}\left (\int \frac {x^7}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {1}{108} x^{10} \sqrt [3]{-1+x^6}+\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}-\frac {5}{108} \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {5}{324} x^4 \sqrt [3]{-1+x^6}-\frac {1}{108} x^{10} \sqrt [3]{-1+x^6}+\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}-\frac {5}{162} \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {5}{324} x^4 \sqrt [3]{-1+x^6}-\frac {1}{108} x^{10} \sqrt [3]{-1+x^6}+\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}-\frac {5}{162} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {5}{324} x^4 \sqrt [3]{-1+x^6}-\frac {1}{108} x^{10} \sqrt [3]{-1+x^6}+\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}-\frac {5}{486} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {5}{486} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {5}{324} x^4 \sqrt [3]{-1+x^6}-\frac {1}{108} x^{10} \sqrt [3]{-1+x^6}+\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}+\frac {5}{486} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {5}{972} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {5}{324} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {5}{324} x^4 \sqrt [3]{-1+x^6}-\frac {1}{108} x^{10} \sqrt [3]{-1+x^6}+\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}+\frac {5}{486} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {5}{972} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {5}{324} x^4 \sqrt [3]{-1+x^6}-\frac {1}{108} x^{10} \sqrt [3]{-1+x^6}+\frac {1}{18} x^{16} \sqrt [3]{-1+x^6}+\frac {5 \tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{162 \sqrt {3}}+\frac {5}{486} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {5}{972} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 67, normalized size = 0.57 \begin {gather*} \frac {x^4 \sqrt [3]{x^6-1} \left (5 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^6\right )+\sqrt [3]{1-x^6} \left (6 x^{12}-x^6-5\right )\right )}{108 \sqrt [3]{1-x^6}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.57, size = 117, normalized size = 1.00 \begin {gather*} \frac {5}{486} \log \left (\sqrt [3]{x^6-1}-x^2\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6-1}+x^2}\right )}{162 \sqrt {3}}-\frac {5}{972} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right )+\frac {1}{324} \sqrt [3]{x^6-1} \left (18 x^{16}-3 x^{10}-5 x^4\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 {5}{486} \, \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}{324} \, {\left (18 \, x^{16} - 3 \, x^{10} - 5 \, x^{4}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {5}{972} \, \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 {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{15}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 58, normalized size = 0.50 \begin {gather*} \frac {x^{4} \left (18 x^{12}-3 x^{6}-5\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{324}-\frac {5 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {2}{3}} x^{4} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{324 \mathrm {signum}\left (x^{6}-1\right )^{\frac {2}{3}}} \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 {5}{486} \, \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 {10 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {13 \, {\left (x^{6} - 1\right )}^{\frac {4}{3}}}{x^{8}} - \frac {5 \, {\left (x^{6} - 1\right )}^{\frac {7}{3}}}{x^{14}}}{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 {5}{972} \, \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 {5}{486} \, \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 x^{15}\,{\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 = 1.91, size = 36, normalized size = 0.31 \begin {gather*} - \frac {x^{16} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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