Optimal. Leaf size=122 \[ -\log \left (\sqrt [3]{x^3+x^2-x-1}-x-1\right )+\frac {1}{2} \log \left (x^2+\left (x^3+x^2-x-1\right )^{2/3}+(x+1) \sqrt [3]{x^3+x^2-x-1}+2 x+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^3+x^2-x-1}}{\sqrt [3]{x^3+x^2-x-1}+2 x+2}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 167, normalized size of antiderivative = 1.37, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2067, 2064, 60} \begin {gather*} -\frac {3 (-x-1)^{2/3} \sqrt [3]{x-1} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{-x-1}}+1\right )}{2 \sqrt [3]{x^3+x^2-x-1}}-\frac {(-x-1)^{2/3} \sqrt [3]{x-1} \log \left (-\frac {8}{3} (x+1)\right )}{2 \sqrt [3]{x^3+x^2-x-1}}-\frac {\sqrt {3} (-x-1)^{2/3} \sqrt [3]{x-1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{-x-1}}\right )}{\sqrt [3]{x^3+x^2-x-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 60
Rule 2064
Rule 2067
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{-1-x+x^2+x^3}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\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 {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 )}{3 \sqrt [3]{-1-x+x^2+x^3}}\\ &=-\frac {\sqrt {3} (-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]{-1-x+x^2+x^3}}-\frac {3 (-1-x)^{2/3} \sqrt [3]{-1+x} \log \left (1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{-1-x}}\right )}{2 \sqrt [3]{-1-x+x^2+x^3}}-\frac {(-1-x)^{2/3} \sqrt [3]{-1+x} \log (1+x)}{2 \sqrt [3]{-1-x+x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 48, normalized size = 0.39 \begin {gather*} \frac {3 \left ((x-1) (x+1)^2\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {1-x}{2}\right )}{2\ 2^{2/3} (x+1)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 122, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-1-x+x^2+x^3}}{2+2 x+\sqrt [3]{-1-x+x^2+x^3}}\right )-\log \left (-1-x+\sqrt [3]{-1-x+x^2+x^3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(1+x) \sqrt [3]{-1-x+x^2+x^3}+\left (-1-x+x^2+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 120, normalized size = 0.98 \begin {gather*} -\sqrt {3} \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 ) + \frac {1}{2} \, \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 ) - \log \left (-\frac {x - {\left (x^{3} + x^{2} - x - 1\right )}^{\frac {1}{3}} + 1}{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 {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 = 0.44, size = 352, normalized size = 2.89
method | result | size |
trager | \(-\ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-1}{1+x}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+3 x \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+2 x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+3 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+4 x +2}{1+x}\right )\) | \(352\) |
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 {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 {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 {1}{\sqrt [3]{x^{3} + x^{2} - x - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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