Optimal. Leaf size=109 \[ \frac {\sqrt [3]{1-x^3} \log \left (\sqrt [3]{1-x^3}+x\right )}{2 \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}}-\frac {\sqrt [3]{1-x^3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {713, 239} \begin {gather*} \frac {\sqrt [3]{1-x^3} \log \left (\sqrt [3]{1-x^3}+x\right )}{2 \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}}-\frac {\sqrt [3]{1-x^3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 713
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{1+x+x^2}} \, dx &=\frac {\sqrt [3]{1-x^3} \int \frac {1}{\sqrt [3]{1-x^3}} \, dx}{\sqrt [3]{1-x} \sqrt [3]{1+x+x^2}}\\ &=-\frac {\sqrt [3]{1-x^3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-x} \sqrt [3]{1+x+x^2}}+\frac {\sqrt [3]{1-x^3} \log \left (x+\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{1-x} \sqrt [3]{1+x+x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 132, normalized size = 1.21 \begin {gather*} -\frac {3 (1-x)^{2/3} \sqrt [3]{\frac {-2 i x+\sqrt {3}-i}{\sqrt {3}-3 i}} \sqrt [3]{\frac {2 i x+\sqrt {3}+i}{\sqrt {3}+3 i}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {2 i (x-1)}{3 i+\sqrt {3}},\frac {2 i (x-1)}{-3 i+\sqrt {3}}\right )}{2 \sqrt [3]{x^2+x+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 13.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{1+x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.60, size = 115, normalized size = 1.06 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} x^{2} {\left (-x + 1\right )}^{\frac {1}{3}} + 2 \, \sqrt {3} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} x {\left (-x + 1\right )}^{\frac {2}{3}} - \sqrt {3} {\left (x^{3} - 1\right )}}{9 \, x^{3} - 1}\right ) + \frac {1}{6} \, \log \left (3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} x^{2} {\left (-x + 1\right )}^{\frac {1}{3}} + 3 \, {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} x {\left (-x + 1\right )}^{\frac {2}{3}} + 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^{2} + x + 1\right )}^{\frac {1}{3}} {\left (-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 [F] time = 2.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-x +1\right )^{\frac {1}{3}} \left (x^{2}+x +1\right )^{\frac {1}{3}}}\, dx \end {gather*}
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^{2} + x + 1\right )}^{\frac {1}{3}} {\left (-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 (1-x\right )}^{1/3}\,{\left (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]{1 - x} \sqrt [3]{x^{2} + x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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