Integrand size = 19, antiderivative size = 132 \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {399, 245, 384} \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\frac {\arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right ) \]
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Rule 245
Rule 384
Rule 399
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx-\int \frac {1}{\sqrt [3]{1-x^3}} \, dx \\ & = \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 {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}+\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 ) \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.55 \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\frac {1}{6} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )-2\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2 \log \left (x+\sqrt [3]{1-x^3}\right )+2\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+\log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-2^{2/3} \log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \]
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Time = 1.85 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.36
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{3}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )}{3}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right )}{3}+\frac {\ln \left (\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}}{x^{2}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}\) | \(179\) |
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none
Time = 0.22 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.45 \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=-\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \cdot 4^{\frac {1}{3}} \log \left (\frac {4^{\frac {2}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3} + 1} \,d x } \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3} + 1} \,d x } \]
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Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{x^3+1} \,d x \]
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