Optimal. Leaf size=84 \[ 3 \sqrt [3]{1-x}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac {\log (x+1)}{2^{2/3}}-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x}+1}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {50, 57, 617, 204, 31} \begin {gather*} 3 \sqrt [3]{1-x}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac {\log (x+1)}{2^{2/3}}-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x}+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1-x}}{1+x} \, dx &=3 \sqrt [3]{1-x}+2 \int \frac {1}{(1-x)^{2/3} (1+x)} \, dx\\ &=3 \sqrt [3]{1-x}-\frac {\log (1+x)}{2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x}\right )}{\sqrt [3]{2}}\\ &=3 \sqrt [3]{1-x}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac {\log (1+x)}{2^{2/3}}+\left (3 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x}\right )\\ &=3 \sqrt [3]{1-x}-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x}}{\sqrt {3}}\right )+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac {\log (1+x)}{2^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 104, normalized size = 1.24 \begin {gather*} 3 \sqrt [3]{1-x}+\sqrt [3]{2} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )-\frac {\log \left ((1-x)^{2/3}+\sqrt [3]{2-2 x}+2^{2/3}\right )}{2^{2/3}}-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x}+1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 115, normalized size = 1.37 \begin {gather*} 3 \sqrt [3]{1-x}+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{1-x}-2\right )-\frac {\log \left (\sqrt [3]{2} (1-x)^{2/3}+2^{2/3} \sqrt [3]{1-x}+2\right )}{2^{2/3}}-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 86, normalized size = 1.02 \begin {gather*} -\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} 2^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-x + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 87, normalized size = 1.04 \begin {gather*} -\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {1}{3}} \right |}\right ) + 3 \, {\left (-x + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 1.00 \begin {gather*} -2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x +1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )+2^{\frac {1}{3}} \ln \left (\left (-x +1\right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )-\frac {2^{\frac {1}{3}} \ln \left (\left (-x +1\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (-x +1\right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{2}+3 \left (-x +1\right )^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 86, normalized size = 1.02 \begin {gather*} -\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-x + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 104, normalized size = 1.24 \begin {gather*} 2^{1/3}\,\ln \left (18\,{\left (1-x\right )}^{1/3}-18\,2^{1/3}\right )+3\,{\left (1-x\right )}^{1/3}+\frac {2^{1/3}\,\ln \left (18\,{\left (1-x\right )}^{1/3}-9\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {2^{1/3}\,\ln \left (18\,{\left (1-x\right )}^{1/3}+9\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.26, size = 170, normalized size = 2.02 \begin {gather*} \frac {4 \sqrt [3]{-1} \sqrt [3]{x - 1} \Gamma \left (\frac {4}{3}\right )}{\Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{-2} e^{- \frac {i \pi }{3}} \log {\left (- \frac {2^{\frac {2}{3}} \sqrt [3]{x - 1} e^{\frac {i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} - \frac {4 \sqrt [3]{-2} \log {\left (- \frac {2^{\frac {2}{3}} \sqrt [3]{x - 1} e^{i \pi }}{2} + 1 \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{-2} e^{\frac {i \pi }{3}} \log {\left (- \frac {2^{\frac {2}{3}} \sqrt [3]{x - 1} e^{\frac {5 i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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