Integrand size = 17, antiderivative size = 97 \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((1-x) (1+x)^2\right )}{4 \sqrt [3]{2}}+\frac {3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
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Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2174} \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}+\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac {\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]
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Rule 2174
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((1-x) (1+x)^2\right )}{4 \sqrt [3]{2}}+\frac {3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \\ \end{align*}
Time = 1.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )+2 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )-\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2-2 (-1+x) \sqrt [3]{2-2 x^3}+4 \left (1-x^3\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.01 (sec) , antiderivative size = 769, normalized size of antiderivative = 7.93
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (71) = 142\).
Time = 1.93 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.10 \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (13 \, x^{6} + 2 \, x^{5} + 19 \, x^{4} - 4 \, x^{3} + 19 \, x^{2} + 2 \, x + 13\right )} - 4 \, \sqrt {2} {\left (5 \, x^{5} - 5 \, x^{4} + 6 \, x^{3} - 6 \, x^{2} + 5 \, x - 5\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 16 \cdot 2^{\frac {1}{6}} {\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (3 \, x^{6} - 18 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 3 \, x^{2} - 18 \, x + 3\right )}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {4 \cdot 2^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 1\right )} + 2^{\frac {1}{3}} {\left (5 \, x^{4} + 6 \, x^{2} + 5\right )} - 2 \, {\left (3 \, x^{3} - x^{2} + x - 3\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} - 2 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 4 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 2 \, x + 1}\right ) \]
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\[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\int \frac {1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right )}\, dx \]
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\[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\int { \frac {1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
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\[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\int { \frac {1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(1+x) \sqrt [3]{1-x^3}} \, dx=\int \frac {1}{{\left (1-x^3\right )}^{1/3}\,\left (x+1\right )} \,d x \]
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