Optimal. Leaf size=103 \[ -\frac {1}{9} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{3 \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (2 x^3-3\right )}{6 x^2}+\frac {1}{18} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {453, 195, 239} \begin {gather*} -\frac {1}{6} x \left (x^3-1\right )^{2/3}-\frac {1}{6} \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 )}{3 \sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 239
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^3} \, dx &=\frac {\left (-1+x^3\right )^{5/3}}{2 x^2}-\frac {1}{2} \int \left (-1+x^3\right )^{2/3} \, dx\\ &=-\frac {1}{6} x \left (-1+x^3\right )^{2/3}+\frac {\left (-1+x^3\right )^{5/3}}{2 x^2}+\frac {1}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=-\frac {1}{6} x \left (-1+x^3\right )^{2/3}+\frac {\left (-1+x^3\right )^{5/3}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 50, normalized size = 0.49 \begin {gather*} \frac {\left (x^3-1\right )^{2/3} \left (-\frac {x^3 \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};x^3\right )}{\left (1-x^3\right )^{2/3}}+x^3-1\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 103, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-3+2 x^3\right )}{6 x^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{18} \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 = 0.98, size = 112, normalized size = 1.09 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \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 ) - x^{2} \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 (2 \, x^{3} - 3\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{18 \, x^{2}} \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^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.30, size = 56, normalized size = 0.54
method | result | size |
risch | \(\frac {2 x^{6}-5 x^{3}+3}{6 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(56\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2}}\) | \(63\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (2 x^{3}-3\right )}{6 x^{2}}+\frac {4 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (182271728 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}+775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+730283524 \RootOf \left (16 \textit {\_Z}^{2}-4 \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}-1458173824 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}+1858712316 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+476369215\right )}{9}+\frac {\ln \left (182271728 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \RootOf \left (16 \textit {\_Z}^{2}-4 \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}-1458173824 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+849911430\right )}{9}-\frac {4 \ln \left (182271728 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \RootOf \left (16 \textit {\_Z}^{2}-4 \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}-1458173824 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+849911430\right ) \RootOf \left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{9}\) | \(457\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 106, normalized size = 1.03 \begin {gather*} -\frac {1}{9} \, \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 {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} + \frac {1}{18} \, \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}{9} \, \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^3+1\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.26, size = 70, normalized size = 0.68 \begin {gather*} - \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\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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