Integrand size = 23, antiderivative size = 119 \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2}}-\frac {\log \left (1+\frac {2^{2/3} (1+x)^2}{\left (1+x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{2}} \]
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Leaf count is larger than twice the leaf count of optimal. \(357\) vs. \(2(119)=238\).
Time = 0.20 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2183, 384, 502, 2174, 206, 31, 648, 631, 210, 642, 455, 57} \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {2^{2/3} \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1-x^3\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{x^3+1}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{x^3+1}+x+1\right )}{2 \sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (x+1)\right )}{6 \sqrt [3]{2}} \]
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Rule 31
Rule 57
Rule 206
Rule 210
Rule 384
Rule 455
Rule 502
Rule 631
Rule 642
Rule 648
Rule 2174
Rule 2183
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}}-\frac {2 x}{\left (1-x^3\right ) \sqrt [3]{1+x^3}}+\frac {x^2}{\left (1-x^3\right ) \sqrt [3]{1+x^3}}\right ) \, dx \\ & = -\left (2 \int \frac {x}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx\right )+\int \frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx+\int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {2}{3} \int \frac {1}{(1-x) \sqrt [3]{1+x^3}} \, dx+2 \text {Subst}\left (\int \frac {1}{1+2 x^3} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right ) \\ & = \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 {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{6 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {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 {2}{3} \text {Subst}\left (\int \frac {2-\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 {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}} \\ & = \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 {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{6 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}-\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 \sqrt [3]{2}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}+\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 ) \\ & = \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 {\arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{6 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{3 \sqrt [3]{2}}-\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 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}+2^{2/3} \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 {\arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {2^{2/3} \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 )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{6 \sqrt [3]{2}}+\frac {\log \left (1-x^3\right )}{3 \sqrt [3]{2}}-\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 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )-\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{2 \sqrt [3]{2}} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.17 \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+x^3}}{-2 \sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{1+x^3}}\right )+2 \log \left (\sqrt [3]{2}+\sqrt [3]{2} x+\sqrt [3]{1+x^3}\right )-\log \left (2^{2/3}+2\ 2^{2/3} x+2^{2/3} x^2-\sqrt [3]{2} (1+x) \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.40 (sec) , antiderivative size = 714, normalized size of antiderivative = 6.00
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (93) = 186\).
Time = 5.27 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.25 \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{6} + 7 \, x^{5} + 10 \, x^{4} + 7 \, x^{3} + 10 \, x^{2} + 7 \, x + 1\right )} - 4 \, \sqrt {2} {\left (x^{5} + x^{4} - 3 \, x^{3} - 3 \, x^{2} + x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 4 \cdot 2^{\frac {1}{6}} {\left (x^{4} + 4 \, x^{3} + 5 \, x^{2} + 4 \, x + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (3 \, x^{6} + 9 \, x^{5} + 6 \, x^{4} + x^{3} + 6 \, x^{2} + 9 \, x + 3\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3 \, x + 1\right )} - 2^{\frac {1}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 4 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} + 2 \cdot 2^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 2 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + x + 1}\right ) \]
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\[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=- \int \frac {x}{x^{2} \sqrt [3]{x^{3} + 1} + x \sqrt [3]{x^{3} + 1} + \sqrt [3]{x^{3} + 1}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt [3]{x^{3} + 1} + x \sqrt [3]{x^{3} + 1} + \sqrt [3]{x^{3} + 1}}\right )\, dx \]
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\[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\int { -\frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
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\[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=\int { -\frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1-x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^3}} \, dx=-\int \frac {x-1}{{\left (x^3+1\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]
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