Optimal. Leaf size=123 \[ \frac {\log \left (2^{2/3} \sqrt [3]{x^2-2 x-2}-2\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \left (x^2-2 x-2\right )^{2/3}+2^{2/3} \sqrt [3]{x^2-2 x-2}+2\right )}{4 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-2 x-2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \]
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Rubi [F] time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx &=\int \frac {-1+x}{\left (-4-2 x+x^2\right ) \sqrt [3]{-2-2 x+x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.08, size = 78, normalized size = 0.63 \begin {gather*} \frac {-\log \left (x^2-2 x-4\right )+3 \log \left (\sqrt [3]{2}-\sqrt [3]{x^2-2 x-2}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-2 x-2}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 123, normalized size = 1.00 \begin {gather*} \frac {\log \left (2^{2/3} \sqrt [3]{x^2-2 x-2}-2\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \left (x^2-2 x-2\right )^{2/3}+2^{2/3} \sqrt [3]{x^2-2 x-2}+2\right )}{4 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^2-2 x-2}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 93, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} + 2 \, \sqrt {2} {\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{3}} + {\left (x^{2} - 2 \, x - 2\right )}^{\frac {2}{3}}\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (x^{2} - 2 \, x - 2\right )}^{\frac {1}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 84, normalized size = 0.68 \begin {gather*} \frac {1}{4} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{2} \, x^{2} - x - 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{8} \cdot 2^{\frac {2}{3}} \log \left ({\left (\frac {1}{2} \, x^{2} - x - 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{2} \, x^{2} - x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{4} \cdot 2^{\frac {2}{3}} \log \left ({\left | {\left (\frac {1}{2} \, x^{2} - 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 = 9.90, size = 1042, normalized size = 8.47
result too large to display
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^{2} - 2 \, x - 2\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x - 4\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 109, normalized size = 0.89 \begin {gather*} \frac {2^{2/3}\,\ln \left (\frac {9\,{\left (x^2-2\,x-2\right )}^{1/3}}{4}-\frac {9\,2^{1/3}}{4}\right )}{4}+\frac {2^{2/3}\,\ln \left (\frac {9\,{\left (x^2-2\,x-2\right )}^{1/3}}{4}-\frac {9\,2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{8}-\frac {2^{2/3}\,\ln \left (\frac {9\,{\left (x^2-2\,x-2\right )}^{1/3}}{4}-\frac {9\,2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{8} \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}{\left (x^{2} - 2 x - 4\right ) \sqrt [3]{x^{2} - 2 x - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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