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 = 0.0187458, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {713, 239} \[ \frac{\sqrt [3]{1-x^3} \log \left (\sqrt [3]{1-x^3}+x\right )}{2 \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}}-\frac{\sqrt [3]{1-x^3} \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}} \]
Antiderivative was successfully verified.
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Rule 713
Rule 239
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*}
Mathematica [C] time = 0.0650933, size = 132, normalized size = 1.21 \[ -\frac{3 (1-x)^{2/3} \sqrt [3]{\frac{-2 i x+\sqrt{3}-i}{\sqrt{3}-3 i}} \sqrt [3]{\frac{2 i x+\sqrt{3}+i}{\sqrt{3}+3 i}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{2 i (x-1)}{3 i+\sqrt{3}},\frac{2 i (x-1)}{-3 i+\sqrt{3}}\right )}{2 \sqrt [3]{x^2+x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.691, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{{x}^{2}+x+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} + x + 1\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.56897, size = 339, normalized size = 3.11 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{1 - x} \sqrt [3]{x^{2} + x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} + x + 1\right )}^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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