Integrand size = 19, antiderivative size = 88 \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
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Time = 0.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {384} \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}} \]
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Rule 384
Rubi steps \begin{align*} \text {integral}& = -\frac {\arctan \left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2 \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
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Time = 3.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right )}{3 x}\right )+\ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{6}\) | \(95\) |
trager | \(\text {Expression too large to display}\) | \(780\) |
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Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (67) = 134\).
Time = 1.61 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.88 \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=-\frac {1}{18} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, \sqrt {6} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) + \frac {1}{18} \cdot 2^{\frac {2}{3}} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) \]
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\[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int { \frac {1}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx=\int \frac {1}{{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \]
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