Optimal. Leaf size=105 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )+\frac {\left (x^3-1\right )^{2/3} \left (-x^6-2 x^3+1\right )}{4 x^8} \]
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Rubi [A] time = 0.05, antiderivative size = 94, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1488, 271, 264, 277, 239} \begin {gather*} -\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^3-1\right )^{5/3}}{4 x^8}+\frac {\left (x^3-1\right )^{5/3}}{4 x^5}-\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 264
Rule 271
Rule 277
Rule 1488
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^9}+\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {\left (-1+x^3\right )^{2/3}}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx\right )+2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}+\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {3}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}+\frac {\left (-1+x^3\right )^{5/3}}{4 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 57, normalized size = 0.54 \begin {gather*} \frac {\left (x^3-1\right )^{2/3} \left (\left (1-x^3\right )^{8/3}-2 x^6 \, _2F_1\left (-\frac {2}{3},-\frac {2}{3};\frac {1}{3};x^3\right )\right )}{4 x^8 \left (1-x^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 105, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3-x^6\right )}{4 x^8}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 115, normalized size = 1.10 \begin {gather*} \frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - 2 \, x^{8} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{12 \, x^{8}} \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 {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.65, size = 56, normalized size = 0.53
method | result | size |
risch | \(-\frac {x^{9}+x^{6}-3 x^{3}+1}{4 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(56\) |
meijerg | \(-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2}}-\frac {2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{5}}+\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{8}}\) | \(110\) |
trager | \(-\frac {\left (x^{6}+2 x^{3}-1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{4 x^{8}}+\frac {\ln \left (746584997888 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}-49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-52570840832 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-5972679983104 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-72296025856 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+849911430\right )}{3}-\frac {256 \ln \left (746584997888 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}-49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-52570840832 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-5972679983104 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-72296025856 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+849911430\right ) \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )}{3}+\frac {256 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \ln \left (746584997888 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}+49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +49654493184 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+46738145536 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}-1508552373 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1497160390 x^{3}-5972679983104 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}+118957588224 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+476369215\right )}{3}\) | \(460\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 93, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {{\left (x^{3} - 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 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 {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.58, size = 461, normalized size = 4.39 \begin {gather*} 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) - 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {3 x^{6} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) + \frac {e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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