Optimal. Leaf size=111 \[ -\frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2}-\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 155, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2052, 2011, 59, 2014} \begin {gather*} -\frac {3 \left (x^3-x^2\right )^{2/3}}{2 x^2}-\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 2011
Rule 2014
Rule 2052
Rubi steps
\begin {align*} \int \frac {-1+x}{x \sqrt [3]{-x^2+x^3}} \, dx &=\int \left (\frac {1}{\sqrt [3]{-x^2+x^3}}-\frac {1}{x \sqrt [3]{-x^2+x^3}}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx-\int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx\\ &=-\frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{2 \sqrt [3]{-x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 35, normalized size = 0.32 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{5/3} \, _2F_1\left (\frac {5}{3},\frac {5}{3};\frac {8}{3};1-x\right )}{5 x^{10/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 111, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-x^2+x^3\right )^{2/3}}{2 x^2}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 119, normalized size = 1.07 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, x^{2} \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - x^{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 ) + 3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 74, normalized size = 0.67 \begin {gather*} -\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{2} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + \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 = 0.58, size = 42, normalized size = 0.38
method | result | size |
risch | \(-\frac {3 \left (-1+x \right )}{2 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(42\) |
meijerg | \(\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} \left (1-x \right )^{\frac {2}{3}}}{2 \mathrm {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {2}{3}}}+\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{3}}}\) | \(54\) |
trager | \(-\frac {3 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}+6 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +114 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-18 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -9 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-4 x^{2}+x}{x}\right )-6 \ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x -144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -174 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+138 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right ) \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+\ln \left (\frac {180 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-360 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x -144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-144 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -174 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+138 \RootOf \left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x +15 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+15 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+20 x^{2}-12 x}{x}\right )\) | \(504\) |
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 {x - 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 44, normalized size = 0.40 \begin {gather*} \frac {3\,x\,{\left (1-x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ x\right )}{{\left (x^3-x^2\right )}^{1/3}}-\frac {3\,{\left (x^3-x^2\right )}^{2/3}}{2\,x^2} \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 {x - 1}{x \sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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