Optimal. Leaf size=92 \[ -\frac {\left (x^6-1\right )^{2/3}}{6 x^6}-\frac {1}{9} \log \left (\sqrt [3]{x^6-1}+1\right )+\frac {1}{18} \log \left (\left (x^6-1\right )^{2/3}-\sqrt [3]{x^6-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 0.74, 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 = {266, 47, 56, 618, 204, 31} \begin {gather*} -\frac {\left (x^6-1\right )^{2/3}}{6 x^6}-\frac {1}{6} \log \left (\sqrt [3]{x^6-1}+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\log (x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 56
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (-1+x^6\right )^{2/3}}{x^7} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\left (-1+x^6\right )^{2/3}}{6 x^6}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {\left (-1+x^6\right )^{2/3}}{6 x^6}+\frac {\log (x)}{3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^6}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^6}\right )\\ &=-\frac {\left (-1+x^6\right )^{2/3}}{6 x^6}+\frac {\log (x)}{3}-\frac {1}{6} \log \left (1+\sqrt [3]{-1+x^6}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^6}\right )\\ &=-\frac {\left (-1+x^6\right )^{2/3}}{6 x^6}-\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\log (x)}{3}-\frac {1}{6} \log \left (1+\sqrt [3]{-1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.30 \begin {gather*} \frac {1}{10} \left (x^6-1\right )^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};1-x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 92, normalized size = 1.00 \begin {gather*} -\frac {\left (-1+x^6\right )^{2/3}}{6 x^6}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {1}{18} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 80, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{6} \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{6} \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{18 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 69, normalized size = 0.75 \begin {gather*} \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{18} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{9} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.69, size = 96, normalized size = 1.04
method | result | size |
risch | \(-\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{6 x^{6}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(96\) |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{6}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{6} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{18 \pi \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {2}{3}}}\) | \(97\) |
trager | \(-\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{6 x^{6}}+\frac {\ln \left (\frac {-6164580662439356682428416 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )^{2} x^{6}-5852076754897126485048384 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right ) x^{6}+116115113033469041646202 x^{6}+4369107848502849252902784 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-4369107848502849252902784 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+394533162396118827675418624 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )^{2}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}-1795472813936507429525632 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )-114242288629703411942231}{x^{6}}\right )}{9}-\frac {64 \ln \left (\frac {-6164580662439356682428416 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )^{2} x^{6}-5852076754897126485048384 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right ) x^{6}+116115113033469041646202 x^{6}+4369107848502849252902784 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )-210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-4369107848502849252902784 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+394533162396118827675418624 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )^{2}+210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}-1795472813936507429525632 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )-114242288629703411942231}{x^{6}}\right ) \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )}{9}+\frac {64 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right ) \ln \left (-\frac {6164580662439356682428416 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )^{2} x^{6}-6044719900598356381374272 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right ) x^{6}-23171389162410581752275 x^{6}+4369107848502849252902784 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )+142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}-4369107848502849252902784 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}-394533162396118827675418624 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )^{2}-142296551347461340528569 \left (x^{6}-1\right )^{\frac {1}{3}}+10533688510942205935331200 \RootOf \left (4096 \textit {\_Z}^{2}-64 \textit {\_Z} +1\right )+45974978496846392365625}{x^{6}}\right )}{9}\) | \(439\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 68, normalized size = 0.74 \begin {gather*} \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{6}} + \frac {1}{18} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{9} \, \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 = 0.90, size = 92, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (\frac {{\left (x^6-1\right )}^{1/3}}{9}+\frac {1}{9}\right )}{9}-\ln \left (9\,{\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{9}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\ln \left (9\,{\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{9}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^6-1\right )}^{2/3}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.12, size = 34, normalized size = 0.37 \begin {gather*} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 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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