Integrand size = 22, antiderivative size = 234 \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=-\frac {\left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {2 x^2 \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}} \]
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Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {2183, 386, 384, 480, 21, 371, 455, 43, 57, 631, 210, 31} \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {\left (1-x^3\right )^{2/3} x}{3 \left (x^3+1\right )}-\frac {\left (1-x^3\right )^{2/3}}{3 \left (x^3+1\right )}-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}+\frac {2 \left (1-x^3\right )^{2/3} x^2}{3 \left (x^3+1\right )} \]
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Rule 21
Rule 31
Rule 43
Rule 57
Rule 210
Rule 371
Rule 384
Rule 386
Rule 455
Rule 480
Rule 631
Rule 2183
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {2 x \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {x^2 \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2}\right ) \, dx \\ & = 2 \int \frac {x \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\int \frac {x^2 \left (1-x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx \\ & = \frac {x \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {2 x^2 \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {1}{3} \text {Subst}\left (\int \frac {(1-x)^{2/3}}{(1+x)^2} \, dx,x,x^3\right )+\frac {2}{3} \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx-\frac {2}{3} \int \frac {x \left (-1-x^3\right )}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {2 x^2 \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )+\frac {2}{3} \int \frac {x}{\sqrt [3]{1-x^3}} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {2 x^2 \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}} \\ & = -\frac {\left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {2 x^2 \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right ) \\ & = -\frac {\left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {2 x^2 \left (1-x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}} \\ \end{align*}
\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]
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\[\int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{\left (x^{2}-x +1\right )^{2}}d x\]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x^{2} - x + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} - x + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{{\left (x^2-x+1\right )}^2} \,d x \]
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