Integrand size = 19, antiderivative size = 272 \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}} \]
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Time = 0.10 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {420, 493, 298, 31, 648, 631, 210, 642} \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3}} \]
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Rule 31
Rule 210
Rule 298
Rule 420
Rule 493
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\left (9 \text {Subst}\left (\int \frac {x}{\left (4-x^3\right ) \left (1+2 x^3\right )} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {x}{1+2 x^3} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )\right )-\text {Subst}\left (\int \frac {x}{4-x^3} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {\text {Subst}\left (\int \frac {2^{2/3}-x}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right ) \\ & = \frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )-\frac {\text {Subst}\left (\int \frac {2^{2/3}+2 x}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}} \\ & = \frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{2^{2/3}}-\sqrt [3]{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right ) \\ & = \frac {\sqrt [3]{2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (2 \sqrt [3]{2}+\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}} \\ \end{align*}
Time = 1.91 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{2}-\sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{-2 \sqrt [3]{2}+2 \sqrt [3]{2} x+\sqrt [3]{1-x^3}}\right )-4 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )-2 \log \left (-\sqrt [3]{2}+\sqrt [3]{2} x+2 \sqrt [3]{1-x^3}\right )+2 \log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2+(-1+x) \sqrt [3]{2-2 x^3}+\left (1-x^3\right )^{2/3}\right )+\log \left (2^{2/3}-2\ 2^{2/3} x+2^{2/3} x^2-2 (-1+x) \sqrt [3]{2-2 x^3}+4 \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.60 (sec) , antiderivative size = 1147, normalized size of antiderivative = 4.22
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none
Time = 1.54 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\frac {1}{18} \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (-\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 24 \, \sqrt {3} 2^{\frac {1}{3}} {\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {3} {\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )}}{3 \, {\left (x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1\right )}}\right ) + \frac {1}{18} \cdot 2^{\frac {1}{3}} \log \left (-\frac {12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {1}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{8} - 4 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 6 \, {\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) \]
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\[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\int \frac {\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{\left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 1} \,d x } \]
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\[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 1} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{1-x^3}}{1+x^3} \, dx=\int \frac {{\left (1-x^3\right )}^{1/3}}{x^3+1} \,d x \]
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