Optimal. Leaf size=141 \[ \frac {(x-1)^{2/3} \sqrt [3]{x+1} \left (\sqrt [3]{x-1} (x+1)^{2/3}+\frac {2}{3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x+1}\right )-\frac {1}{3} \log \left ((x-1)^{2/3}+\sqrt [3]{x+1} \sqrt [3]{x-1}+(x+1)^{2/3}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x+1}}{2 \sqrt [3]{x-1}+\sqrt [3]{x+1}}\right )}{\sqrt {3}}\right )}{\sqrt [3]{(x-1)^2 (x+1)}} \]
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Rubi [A] time = 0.20, antiderivative size = 214, normalized size of antiderivative = 1.52, 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 = {2081, 2077, 21, 50, 60} \begin {gather*} -\frac {(1-x) (x+1)}{\sqrt [3]{x^3-x^2-x+1}}+\frac {(3-3 x)^{2/3} \sqrt [3]{x+1} \log \left (-\frac {8}{3} (x-1)\right )}{3\ 3^{2/3} \sqrt [3]{x^3-x^2-x+1}}+\frac {(3-3 x)^{2/3} \sqrt [3]{x+1} \log \left (\frac {\sqrt [3]{3} \sqrt [3]{x+1}}{\sqrt [3]{3-3 x}}+1\right )}{3^{2/3} \sqrt [3]{x^3-x^2-x+1}}+\frac {2 (3-3 x)^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x+1}}{\sqrt [6]{3} \sqrt [3]{3-3 x}}\right )}{3 \sqrt [6]{3} \sqrt [3]{x^3-x^2-x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 60
Rule 2077
Rule 2081
Rubi steps
\begin {align*} \int \frac {-1+x}{\sqrt [3]{1-x-x^2+x^3}} \, dx &=\operatorname {Subst}\left (\int \frac {-\frac {2}{3}+x}{\sqrt [3]{\frac {16}{27}-\frac {4 x}{3}+x^3}} \, dx,x,-\frac {1}{3}+x\right )\\ &=\frac {\left (4\ 2^{2/3} (1-x)^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {2}{3}+x}{\left (\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,-\frac {1}{3}+x\right )}{3 \sqrt [3]{1-x-x^2+x^3}}\\ &=-\frac {\left ((1-x)^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{\frac {16}{9}-\frac {8 x}{3}}}{\sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,-\frac {1}{3}+x\right )}{\sqrt [3]{2} \sqrt [3]{1-x-x^2+x^3}}\\ &=-\frac {(1-x) (1+x)}{\sqrt [3]{1-x-x^2+x^3}}-\frac {\left (8\ 2^{2/3} (1-x)^{2/3} \sqrt [3]{1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {16}{9}-\frac {8 x}{3}\right )^{2/3} \sqrt [3]{\frac {16}{9}+\frac {4 x}{3}}} \, dx,x,-\frac {1}{3}+x\right )}{9 \sqrt [3]{1-x-x^2+x^3}}\\ &=-\frac {(1-x) (1+x)}{\sqrt [3]{1-x-x^2+x^3}}+\frac {2 (1-x)^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{\sqrt {3} \sqrt [3]{1-x-x^2+x^3}}+\frac {(1-x)^{2/3} \sqrt [3]{1+x} \log (1-x)}{3 \sqrt [3]{1-x-x^2+x^3}}+\frac {(1-x)^{2/3} \sqrt [3]{1+x} \log \left (\frac {3 \left (\sqrt [3]{1-x}+\sqrt [3]{1+x}\right )}{\sqrt [3]{1-x}}\right )}{\sqrt [3]{1-x-x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 48, normalized size = 0.34 \begin {gather*} \frac {3 \left ((x-1)^2 (x+1)\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {4}{3};\frac {7}{3};\frac {1-x}{2}\right )}{4 \sqrt [3]{2} (x+1)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.04, size = 141, normalized size = 1.00 \begin {gather*} \frac {(-1+x)^{2/3} \sqrt [3]{1+x} \left (\sqrt [3]{-1+x} (1+x)^{2/3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )}{\sqrt {3}}+\frac {2}{3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-\frac {1}{3} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{\sqrt [3]{(-1+x)^2 (1+x)}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 161, normalized size = 1.14 \begin {gather*} \frac {2 \, \sqrt {3} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {3} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}{3 \, {\left (x - 1\right )}}\right ) - {\left (x - 1\right )} \log \left (\frac {x^{2} + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 2 \, x + {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}} + 1}{x^{2} - 2 \, x + 1}\right ) + 2 \, {\left (x - 1\right )} \log \left (-\frac {x - {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}} - 1}{x - 1}\right ) + 3 \, {\left (x^{3} - x^{2} - x + 1\right )}^{\frac {2}{3}}}{3 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{{\left (x^{3} - x^{2} - x + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.82, size = 361, normalized size = 2.56
method | result | size |
risch | \(\frac {\left (-1+x \right ) \left (1+x \right )}{\left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}}}+\frac {2 \ln \left (-\frac {-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{-1+x}\right )}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4 x -2}{-1+x}\right )}{3}\) | \(361\) |
trager | \(\frac {\left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}}{-1+x}+2 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +2 x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{-1+x}\right )-\frac {2 \ln \left (\frac {-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2 x +1}{-1+x}\right )}{3}-2 \ln \left (\frac {-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x +3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +x^{2}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2 x +1}{-1+x}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(658\) |
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} - x + 1\right )}^{\frac {1}{3}}}\,{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 = -0.01 \begin {gather*} \int \frac {x-1}{{\left (x^3-x^2-x+1\right )}^{1/3}} \,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 {x - 1}{\sqrt [3]{\left (x - 1\right )^{2} \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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