Optimal. Leaf size=86 \[ \frac {1}{2} \left (-1+x^3\right )^{2/3}+\frac {\text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 0.72, 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 = {272, 52, 58,
632, 210, 31} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \left (x^3-1\right )^{2/3}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 58
Rule 210
Rule 272
Rule 632
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3}}{x} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(-1+x)^{2/3}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (-1+x^3\right )^{2/3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (-1+x^3\right )^{2/3}-\frac {\log (x)}{2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=\frac {1}{2} \left (-1+x^3\right )^{2/3}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=\frac {1}{2} \left (-1+x^3\right )^{2/3}-\frac {\tan ^{-1}\left (\frac {-1+2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 82, normalized size = 0.95 \begin {gather*} \frac {1}{6} \left (3 \left (-1+x^3\right )^{2/3}+2 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [3]{-1+x^3}\right )-\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.70, size = 84, normalized size = 0.98
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}}}\) | \(84\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {1438 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+9855 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-5502 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-11504 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+5502 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-17006 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right )}{3}-\frac {\ln \left (-\frac {1438 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-6979 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}-11504 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5502 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-6002 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {\ln \left (-\frac {1438 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-6979 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+5502 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}-11504 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5502 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-6002 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right )}{3}\) | \(358\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 65, normalized size = 0.76 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{2} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (x^{3} - 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 = 0.43, size = 67, normalized size = 0.78 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \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 = 0.53, size = 37, normalized size = 0.43 \begin {gather*} - \frac {x^{2} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 66, normalized size = 0.77 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{2} \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.85, size = 83, normalized size = 0.97 \begin {gather*} \frac {\ln \left ({\left (x^3-1\right )}^{1/3}+1\right )}{3}+\frac {{\left (x^3-1\right )}^{2/3}}{2}+\ln \left (9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^3-1\right )}^{1/3}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+{\left (x^3-1\right )}^{1/3}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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