Optimal. Leaf size=383 \[ \frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )-\frac {\log \left (x^3+1\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left ((1-x) (x+1)^2\right )}{6 \sqrt [3]{2}} \]
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Rubi [F] time = 0.40, 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 = {} \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=\int \left (-\frac {2 \left (1-x^3\right )^{2/3}}{3 (-1-x)}+\frac {\left (-1-(-1)^{2/3}\right ) \left (1-x^3\right )^{2/3}}{3 \left (-1+\sqrt [3]{-1} x\right )}+\frac {\left (-1+\sqrt [3]{-1}\right ) \left (1-x^3\right )^{2/3}}{3 \left (-1-(-1)^{2/3} x\right )}\right ) \, dx\\ &=-\left (\frac {2}{3} \int \frac {\left (1-x^3\right )^{2/3}}{-1-x} \, dx\right )+\frac {1}{3} \left (-1+\sqrt [3]{-1}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-1-(-1)^{2/3} x} \, dx+\frac {1}{3} \left (-1-(-1)^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-1+\sqrt [3]{-1} x} \, dx\\ \end {align*}
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Mathematica [C] time = 0.16, size = 138, normalized size = 0.36 \[ -\frac {4 \left (1-x^3\right )^{2/3} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac {4}{3};-\frac {2}{3},2;\frac {7}{3};x^3,-x^3\right )+2 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )\right )}-\frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 11.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}}{x^{3} + 1}, x\right ) \]
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 \[ \int -\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}}{x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}\, dx \]
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 \[ -\int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}}{x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {{\left (1-x^3\right )}^{2/3}\,\left (x-1\right )}{x^3+1} \,d x \]
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 \[ - \int \left (- \frac {\left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{3} + 1}\right )\, dx - \int \frac {x \left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{3} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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