Optimal. Leaf size=219 \[ -\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{2^{2/3}}+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{2\ 2^{2/3}}-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{2^{2/3}} \]
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Rubi [A] time = 0.08, antiderivative size = 298, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2056, 848, 105, 59, 91} \begin {gather*} \frac {3 \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{2} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{2\ 2^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {3 \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{2 \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [3]{x^3-x^2} \log (x-1)}{2 \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [3]{x^3-x^2} \log (x+1)}{2\ 2^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt {3} \sqrt [3]{x^3-x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt {3} \sqrt [3]{x^3-x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt [3]{x-1} x^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 105
Rule 848
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x^2+x^3}}{-1+x^2} \, dx &=\frac {\sqrt [3]{-x^2+x^3} \int \frac {\sqrt [3]{-1+x} x^{2/3}}{-1+x^2} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\sqrt [3]{-x^2+x^3} \int \frac {x^{2/3}}{(-1+x)^{2/3} (1+x)} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\sqrt [3]{-x^2+x^3} \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x} (1+x)} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt {3} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{2^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {3 \sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\sqrt [3]{2} \sqrt [3]{x}\right )}{2\ 2^{2/3} \sqrt [3]{-1+x} x^{2/3}}-\frac {3 \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log (-1+x)}{2 \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log (1+x)}{2\ 2^{2/3} \sqrt [3]{-1+x} x^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.28 \begin {gather*} \frac {3 \sqrt [3]{(x-1) x^2} \left (2 \sqrt [3]{x} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};1-x\right )-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x-1}{2 x}\right )\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 219, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{2^{2/3}}-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{2^{2/3}}+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 210, normalized size = 0.96 \begin {gather*} -\frac {1}{2} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4^{\frac {1}{3}} x + 4^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + \frac {1}{4} \cdot 4^{\frac {2}{3}} \log \left (-\frac {4^{\frac {2}{3}} x - 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{8} \cdot 4^{\frac {2}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 148, normalized size = 0.68 \begin {gather*} -\frac {1}{2} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) + \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.73, size = 1679, normalized size = 7.67
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1679\) |
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*} \int \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x^{2} - 1}\,{d x} \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 = -0.00 \begin {gather*} \int \frac {{\left (x^3-x^2\right )}^{1/3}}{x^2-1} \,d 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^{2} \left (x - 1\right )}}{\left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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