Optimal. Leaf size=71 \[ 12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 = {43, 52, 60, 632,
210, 31} \begin {gather*} 4 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{3 x-1}}{\sqrt {3}}\right )-\frac {(3 x-1)^{4/3}}{x}+12 \sqrt [3]{3 x-1}+2 \log (x)-6 \log \left (\sqrt [3]{3 x-1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 52
Rule 60
Rule 210
Rule 632
Rubi steps
\begin {align*} \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx &=-\frac {(-1+3 x)^{4/3}}{x}+4 \int \frac {\sqrt [3]{-1+3 x}}{x} \, dx\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}-4 \int \frac {1}{x (-1+3 x)^{2/3}} \, dx\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+2 \log (x)-6 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+3 x}\right )-6 \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right )+12 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+3 x}\right )\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 85, normalized size = 1.20 \begin {gather*} \frac {\sqrt [3]{-1+3 x} (1+9 x)}{x}+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )-4 \log \left (1+\sqrt [3]{-1+3 x}\right )+2 \log \left (1-\sqrt [3]{-1+3 x}+(-1+3 x)^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 109, normalized size = 1.54
method | result | size |
meijerg | \(-\frac {4 \mathrm {signum}\left (x -\frac {1}{3}\right )^{\frac {4}{3}} \left (\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x}+3 \left (2+\frac {\pi \sqrt {3}}{6}-\frac {\ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )-\frac {3 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 3\right ], 3 x \right )}{2}\right )}{3 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x -\frac {1}{3}\right )\right )^{\frac {4}{3}}}\) | \(67\) |
derivativedivides | \(9 \left (3 x -1\right )^{\frac {1}{3}}+\frac {1+\left (3 x -1\right )^{\frac {1}{3}}}{\left (3 x -1\right )^{\frac {2}{3}}-\left (3 x -1\right )^{\frac {1}{3}}+1}+2 \ln \left (\left (3 x -1\right )^{\frac {2}{3}}-\left (3 x -1\right )^{\frac {1}{3}}+1\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (3 x -1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )-\frac {1}{1+\left (3 x -1\right )^{\frac {1}{3}}}-4 \ln \left (1+\left (3 x -1\right )^{\frac {1}{3}}\right )\) | \(109\) |
default | \(9 \left (3 x -1\right )^{\frac {1}{3}}+\frac {1+\left (3 x -1\right )^{\frac {1}{3}}}{\left (3 x -1\right )^{\frac {2}{3}}-\left (3 x -1\right )^{\frac {1}{3}}+1}+2 \ln \left (\left (3 x -1\right )^{\frac {2}{3}}-\left (3 x -1\right )^{\frac {1}{3}}+1\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (3 x -1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )-\frac {1}{1+\left (3 x -1\right )^{\frac {1}{3}}}-4 \ln \left (1+\left (3 x -1\right )^{\frac {1}{3}}\right )\) | \(109\) |
risch | \(\frac {\left (3 x -1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\frac {4 \left (3 x -1\right )^{\frac {2}{3}} \left (-\mathrm {signum}\left (x -\frac {1}{3}\right )\right )^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {\ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+2 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], 3 x \right )\right )}{\left (\left (3 x -1\right )^{2}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x -\frac {1}{3}\right )^{\frac {2}{3}}}+\frac {9 \left (3 x -1\right )^{\frac {2}{3}} \left (-\mathrm {signum}\left (x -\frac {1}{3}\right )\right )^{\frac {2}{3}} x \hypergeom \left (\left [\frac {2}{3}, 1\right ], \left [2\right ], 3 x \right )}{\left (\left (3 x -1\right )^{2}\right )^{\frac {1}{3}} \mathrm {signum}\left (x -\frac {1}{3}\right )^{\frac {2}{3}}}\right ) \left (\left (3 x -1\right )^{2}\right )^{\frac {1}{3}}}{\left (3 x -1\right )^{\frac {2}{3}}}\) | \(146\) |
trager | \(\frac {\left (1+9 x \right ) \left (3 x -1\right )^{\frac {1}{3}}}{x}-4 \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (3 x -1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +\left (3 x -1\right )^{\frac {1}{3}}}{x}\right )+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (3 x -1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (3 x -1\right )^{\frac {1}{3}}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -\left (3 x -1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{x}\right )\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.53, size = 76, normalized size = 1.07 \begin {gather*} -4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + 9 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} + \frac {{\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} + 2 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.41, size = 80, normalized size = 1.13 \begin {gather*} -\frac {4 \, \sqrt {3} x \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (3 \, x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 2 \, x \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) + 4 \, x \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - {\left (9 \, x + 1\right )} {\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.27, size = 541, normalized size = 7.62 \begin {gather*} \frac {189 \cdot \sqrt [3]{3} \left (x - \frac {1}{3}\right )^{\frac {4}{3}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \cdot \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \left (x - \frac {1}{3}\right ) \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {84 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{i \pi } + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \left (x - \frac {1}{3}\right ) e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {28 e^{\frac {i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{i \pi } + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 76, normalized size = 1.07 \begin {gather*} -4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + 9 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} + \frac {{\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} + 2 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, \log \left ({\left (3 \, x - 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 = 0.21, size = 90, normalized size = 1.27 \begin {gather*} 9\,{\left (3\,x-1\right )}^{1/3}-4\,\ln \left (144\,{\left (3\,x-1\right )}^{1/3}+144\right )+\frac {{\left (3\,x-1\right )}^{1/3}}{x}+\ln \left (18-36\,{\left (3\,x-1\right )}^{1/3}+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )-\ln \left (36\,{\left (3\,x-1\right )}^{1/3}-18+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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