Integrand size = 24, antiderivative size = 211 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {3 \left (5+6 x+9 x^2\right ) \left (-x^2+x^3\right )^{2/3}}{40 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.64, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2081, 6857, 129, 491, 597, 12, 384} \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{x^3-x^2}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{x^3-x^2}}+\frac {1-x}{20 x \sqrt [3]{x^3-x^2}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{x^3-x^2}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{x^3-x^2}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{x^3-x^2}}-\frac {17 (1-x)}{40 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{2} \sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{-1}-x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (-x-(-1)^{2/3}\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+1)}{6 \sqrt [3]{2} \sqrt [3]{x^3-x^2}} \]
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Rule 12
Rule 129
Rule 384
Rule 491
Rule 597
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (1+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt [3]{-1+x} x^{11/3}}-\frac {1}{3 \sqrt [3]{-1+x} x^{11/3} \left (-1+\sqrt [3]{-1} x\right )}-\frac {1}{3 \sqrt [3]{-1+x} x^{11/3} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) \sqrt [3]{-1+x} x^{11/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{x^9 \left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{x^9 \sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{x^9 \sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2+6 x^3}{x^6 \left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{8 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {2 \left (5+2 i \sqrt {3}\right )-6 \sqrt [3]{-1} x^3}{x^6 \sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{8 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {2 \left (5-2 i \sqrt {3}\right )+6 (-1)^{2/3} x^3}{x^6 \sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{8 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-34+6 x^3}{x^3 \left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{40 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (5+16 i \sqrt {3}\right )+6 \sqrt [3]{-1} \left (5+2 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{40 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (5-16 i \sqrt {3}\right )-3 \left (1+7 i \sqrt {3}\right ) x^3}{x^3 \sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{40 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int -\frac {80}{\left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{80 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int -\frac {80}{\sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{80 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int -\frac {80}{\sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{80 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5-2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (5+2 i \sqrt {3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{-1+x}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{2} \sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\sqrt [3]{-1}-x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-(-1)^{2/3}-x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (1+x)}{6 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {-45-9 x-27 x^2+81 x^3-20\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} x^{8/3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )+20\ 2^{2/3} \sqrt [3]{-1+x} x^{8/3} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )-10\ 2^{2/3} \sqrt [3]{-1+x} x^{8/3} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )+40 \sqrt [3]{-1+x} x^{8/3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{120 x^2 \sqrt [3]{(-1+x) x^2}} \]
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Time = 14.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(\frac {-10 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{4}+40 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) x^{4}+20 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x^{4}+20 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) x^{4}+81 \left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}} \left (x^{2}+\frac {2}{3} x +\frac {5}{9}\right )}{120 x^{4}}\) | \(176\) |
trager | \(\text {Expression too large to display}\) | \(1758\) |
risch | \(\text {Expression too large to display}\) | \(4518\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.94 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\frac {20 \cdot 2^{\frac {2}{3}} x^{4} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 20 \cdot 2^{\frac {2}{3}} x^{4} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (-i \, \sqrt {3} 2^{\frac {1}{3}} x - 2^{\frac {1}{3}} x\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 20 \, \sqrt {6} 2^{\frac {1}{6}} x^{4} \arctan \left (\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x + 2 \, \sqrt {6} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + 20 \cdot 2^{\frac {2}{3}} x^{4} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 10 \cdot 2^{\frac {2}{3}} x^{4} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x - i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x - i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} - x^{4}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (i \, \sqrt {-3} x + i \, x\right )} + 2^{\frac {1}{3}} {\left (\sqrt {-3} x + x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (\sqrt {-3} x^{4} + x^{4}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {3} 2^{\frac {1}{3}} {\left (-i \, \sqrt {-3} x + i \, x\right )} - 2^{\frac {1}{3}} {\left (\sqrt {-3} x - x\right )}\right )} {\left (-i \, \sqrt {3} + 1\right )}^{\frac {2}{3}} + 8 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 9 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}} {\left (9 \, x^{2} + 6 \, x + 5\right )}}{120 \, x^{4}} \]
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Not integrable
Time = 0.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )} x^{3}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 38.66 (sec) , antiderivative size = 1007, normalized size of antiderivative = 4.77 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 7.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{x^3\,\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]
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