Integrand size = 24, antiderivative size = 383 \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{6 \sqrt [3]{2}}-\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {\log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
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Time = 0.67 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.69, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6857, 2178, 2177, 245, 2174, 371} \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {2^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (1+(-1)^{2/3}\right ) \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\left (1-\sqrt [3]{-1}\right ) \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (1-\sqrt [3]{-1}\right ) \arctan \left (\frac {1-\frac {\sqrt [3]{2} \left (x+\sqrt [3]{-1}\right )}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\left (1+(-1)^{2/3}\right ) \arctan \left (\frac {\frac {(-1)^{2/3} \sqrt [3]{2} \left (\sqrt [3]{-1} x+1\right )}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {1}{6} \left (1+(-1)^{2/3}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {1}{6} \left (1+(-1)^{2/3}\right ) \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {1}{3} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\left (1-\sqrt [3]{-1}\right ) \log \left (-(-2)^{2/3} \sqrt [3]{1-x^3}-(-1)^{2/3} x+1\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{1-x^3}-x+1\right )}{\sqrt [3]{2}}+\frac {\left (1+(-1)^{2/3}\right ) \log \left (\sqrt [3]{-1} 2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{-1} x+1\right )}{2 \sqrt [3]{2}}-\frac {\log \left (-\left ((1-x) (x+1)^2\right )\right )}{3 \sqrt [3]{2}}-\frac {\left (1+(-1)^{2/3}\right ) \log \left (-(-1)^{2/3} \left (x+(-1)^{2/3}\right )^2 \left (\sqrt [3]{-1} x+1\right )\right )}{6 \sqrt [3]{2}}-\frac {\left (1-\sqrt [3]{-1}\right ) \log \left ((-1)^{2/3} \left (x+\sqrt [3]{-1}\right ) \left ((-1)^{2/3} x+1\right )^2\right )}{6 \sqrt [3]{2}} \]
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Rule 245
Rule 371
Rule 2174
Rule 2177
Rule 2178
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \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 \\ & = -\left (\frac {2}{3} \int \frac {1-x}{(-1-x) \sqrt [3]{1-x^3}} \, dx\right )+\frac {2}{3} \int \frac {x}{\sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (1-\sqrt [3]{-1}\right ) \int \frac {x}{\sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (-1+\sqrt [3]{-1}\right ) \int \frac {(-1)^{2/3}-x}{\left (-1+\sqrt [3]{-1} x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )\right ) \int \frac {-\sqrt [3]{-1}-x}{\left (-1-(-1)^{2/3} x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (1+(-1)^{2/3}\right ) \int \frac {x}{\sqrt [3]{1-x^3}} \, dx \\ & = \frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{6} \left (1+(-1)^{2/3}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {2}{3} \int \frac {1}{\sqrt [3]{1-x^3}} \, dx-\frac {4}{3} \int \frac {1}{(-1-x) \sqrt [3]{1-x^3}} \, dx-\frac {1}{3} \left (2 \left (1-\sqrt [3]{-1}\right )\right ) \int \frac {1}{\left (-1-(-1)^{2/3} x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (-1+\sqrt [3]{-1}\right ) \int \frac {1}{\sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )\right ) \int \frac {1}{\sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (2 (-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )\right ) \int \frac {1}{\left (-1+\sqrt [3]{-1} x\right ) \sqrt [3]{1-x^3}} \, dx \\ & = -\frac {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\left (1-\sqrt [3]{-1}\right ) \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{18} \left (3 i+\sqrt {3}\right ) \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )-\frac {\left (1-\sqrt [3]{-1}\right ) \arctan \left (\frac {1-\frac {\sqrt [3]{2} \left (\sqrt [3]{-1}+x\right )}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {(-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \arctan \left (\frac {1+\frac {(-1)^{2/3} \sqrt [3]{2} \left (1+\sqrt [3]{-1} x\right )}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {1}{6} \left (1+(-1)^{2/3}\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left (-\left ((1-x) (1+x)^2\right )\right )}{3 \sqrt [3]{2}}-\frac {(-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \log \left (-(-1)^{2/3} \left ((-1)^{2/3}+x\right )^2 \left (1+\sqrt [3]{-1} x\right )\right )}{6 \sqrt [3]{2}}-\frac {\left (1-\sqrt [3]{-1}\right ) \log \left ((-1)^{2/3} \left (\sqrt [3]{-1}+x\right ) \left (1+(-1)^{2/3} x\right )^2\right )}{6 \sqrt [3]{2}}-\frac {1}{3} \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {1}{6} \left (1-\sqrt [3]{-1}\right ) \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {1}{6} (-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {\left (1-\sqrt [3]{-1}\right ) \log \left (1-(-1)^{2/3} x-(-2)^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1-x-2^{2/3} \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}+\frac {(-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) \log \left (1+\sqrt [3]{-1} x+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 15.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.36 \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {1}{2} x^2 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^3,-x^3\right )-\frac {4 x \left (1-x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {2}{3},1,\frac {4}{3},x^3,-x^3\right )}{\left (1+x^3\right ) \left (-4 \operatorname {AppellF1}\left (\frac {1}{3},-\frac {2}{3},1,\frac {4}{3},x^3,-x^3\right )+x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},-\frac {2}{3},2,\frac {7}{3},x^3,-x^3\right )+2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},x^3,-x^3\right )\right )\right )} \]
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\[\int \frac {\left (1-x \right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}d x\]
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\[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { -\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}}{x^{3} + 1} \,d x } \]
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\[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=- \int \left (- \frac {\left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{3} + 1}\right )\, dx - \int \frac {x \left (1 - x^{3}\right )^{\frac {2}{3}}}{x^{3} + 1}\, dx \]
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\[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { -\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}}{x^{3} + 1} \,d x } \]
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\[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { -\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}}{x^{3} + 1} \,d x } \]
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Timed out. \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\int \frac {{\left (1-x^3\right )}^{2/3}\,\left (x-1\right )}{x^3+1} \,d x \]
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