Optimal. Leaf size=106 \[ -\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {1}{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )-\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 125, normalized size of antiderivative = 1.18, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2045, 2057,
335, 281, 337} \begin {gather*} -\frac {\sqrt {3} x^{2/3} \left (x^2-1\right )^{2/3} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \left (x^3-x\right )^{2/3}}-\frac {3 \sqrt [3]{x^3-x}}{2 x}-\frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 281
Rule 335
Rule 337
Rule 2045
Rule 2057
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{\left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}-\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}+\frac {\left (3 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {\sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{2 \left (-x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \left (-x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 142, normalized size = 1.34 \begin {gather*} -\frac {\left (-1+x^2\right )^{2/3} \left (6 \sqrt [3]{-1+x^2}+2 \sqrt {3} x^{2/3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 x^{2/3} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-x^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{4 \left (x \left (-1+x^2\right )\right )^{2/3}} \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 = 3.23, size = 33, normalized size = 0.31
method | result | size |
meijerg | \(-\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{2}\right )}{2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {2}{3}}}\) | \(33\) |
trager | \(-\frac {3 \left (x^{3}-x \right )^{\frac {1}{3}}}{2 x}-\frac {\ln \left (79344 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}+27072 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-91332 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +57648 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}-317376 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+2256 x \left (x^{3}-x \right )^{\frac {1}{3}}-7060 x^{2}+14052 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+2118\right )}{2}+\frac {\ln \left (-101664 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}-27072 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-64260 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +99804 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+406656 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}-2256 x \left (x^{3}-x \right )^{\frac {1}{3}}-6061 x^{2}-85752 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+3857\right )}{2}-6 \ln \left (-101664 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}-27072 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-64260 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {1}{3}} x +99804 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+406656 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}-2256 x \left (x^{3}-x \right )^{\frac {1}{3}}-6061 x^{2}-85752 \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+3857\right ) \RootOf \left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )\) | \(447\) |
risch | \(\text {Expression too large to display}\) | \(788\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 104, normalized size = 0.98 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + x \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + 6 \, {\left (x^{3} - x\right )}^{\frac {1}{3}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 74, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{2} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-x\right )}^{1/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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