Integrand size = 20, antiderivative size = 45 \[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\frac {x \left (1+x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-x^3\right )}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {727, 251} \[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\frac {x \left (x^3+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \]
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Rule 251
Rule 727
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^3\right )^{2/3} \int \frac {1}{\left (1+x^3\right )^{2/3}} \, dx}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \\ & = \frac {x \left (1+x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.06 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.18 \[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\frac {3 \left (-i+\sqrt {3}+2 i x\right ) \sqrt [3]{1+x} \left (-\frac {3 i+\sqrt {3}+\left (-3 i+\sqrt {3}\right ) x}{-3 i+\sqrt {3}+\left (3 i+\sqrt {3}\right ) x}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {4 i \sqrt {3} (1+x)}{\left (3 i+\sqrt {3}\right ) \left (-i+\sqrt {3}+2 i x\right )}\right )}{\left (-3 i+\sqrt {3}\right ) \left (1-x+x^2\right )^{2/3}} \]
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\[\int \frac {1}{\left (1+x \right )^{\frac {2}{3}} \left (x^{2}-x +1\right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (x + 1\right )^{\frac {2}{3}} \left (x^{2} - x + 1\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx=\int \frac {1}{{\left (x+1\right )}^{2/3}\,{\left (x^2-x+1\right )}^{2/3}} \,d x \]
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