Integrand size = 24, antiderivative size = 257 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=-\frac {\left (-x^2+x^3\right )^{2/3}}{2 (-1+x) x}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Result contains complex when optimal does not.
Time = 3.10 (sec) , antiderivative size = 1933, normalized size of antiderivative = 7.52, number of steps used = 42, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2081, 6857, 21, 49, 52, 61, 129, 490, 596, 544, 245, 384} \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=-\frac {x^3}{2 \sqrt [3]{x^3-x^2}}-\frac {(1-x) x^2}{2 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (2+3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \left (2-3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{-1} \left (2+3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{-1} \left (2-3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{x^3-x^2}}-\frac {5 (1-x) x}{6 \sqrt [3]{x^3-x^2}}-\frac {7 \sqrt [3]{x-1} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) x^{2/3}}{9 \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\left (15 i+19 \sqrt {3}\right ) \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}+\frac {\left (3 i+13 \sqrt {3}\right ) \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}-\frac {\left (3 i-13 \sqrt {3}\right ) \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}+\frac {\left (15 i-19 \sqrt {3}\right ) \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{108 \sqrt [3]{x^3-x^2}}-\frac {4 \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{9 \sqrt {3} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )^{2/3} \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {(-1)^{8/9} \sqrt [3]{x-1} \arctan \left (\frac {\frac {2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right ) x^{2/3}}{2 \sqrt {3} \sqrt [3]{x^3-x^2}}-\frac {7 \sqrt [3]{x-1} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right ) x^{2/3}}{18 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (1+6 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{-1} \left (1-6 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \left (4+3 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{-1} \left (4-3 i \sqrt {3}\right ) \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{36 \sqrt [3]{x^3-x^2}}+\frac {2 \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{x}\right ) x^{2/3}}{9 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{2} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )^{2/3} \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{8/9} \sqrt [3]{x-1} \log \left (\sqrt [3]{x-1}-\sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}\right ) x^{2/3}}{4 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{8/9} \sqrt [3]{x-1} \log \left (\sqrt [3]{-1}-x\right ) x^{2/3}}{12 \sqrt [3]{x^3-x^2}}-\frac {(-1)^{2/3} \left (-1+\sqrt [3]{-1}\right )^{2/3} \sqrt [3]{x-1} \log \left (-x-(-1)^{2/3}\right ) x^{2/3}}{12 \sqrt [3]{x^3-x^2}}-\frac {7 \sqrt [3]{x-1} \log (x) x^{2/3}}{54 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} \log (x+1) x^{2/3}}{12 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} \log \left (x+\sqrt [3]{-1}\right ) x^{2/3}}{12 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x^3-x^2}}+\frac {(-1)^{2/3} \sqrt [3]{-\frac {1}{1+\sqrt [3]{-1}}} \sqrt [3]{x-1} \log \left (x-(-1)^{2/3}\right ) x^{2/3}}{12 \sqrt [3]{x^3-x^2}} \]
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Rule 21
Rule 49
Rule 52
Rule 61
Rule 129
Rule 245
Rule 384
Rule 490
Rule 544
Rule 596
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (-1+x^6\right )} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {x^{7/3}}{2 \sqrt [3]{-1+x} \left (1-x^3\right )}-\frac {x^{7/3}}{2 \sqrt [3]{-1+x} \left (1+x^3\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1-x^3\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1+x^3\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {x^{7/3}}{3 (-1-x) \sqrt [3]{-1+x}}-\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )}-\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (\frac {x^{7/3}}{3 (1-x) \sqrt [3]{-1+x}}+\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )}+\frac {x^{7/3}}{3 \sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{2 \sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{(-1-x) \sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{(1-x) \sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{7/3}}{(-1+x)^{4/3}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^9}{\left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{-x^2+x^3}} \\ & = \frac {(1-x) x^2}{12 \sqrt [3]{-x^2+x^3}}-\frac {x^3}{2 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \left (4-2 x^3\right )}{\left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{12 \sqrt [3]{-x^2+x^3}}+\frac {\left (7 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \left (4-2 \left (3-2 (-1)^{2/3}\right ) x^3\right )}{\sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{12 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \left (-4+2 \left (3+2 (-1)^{2/3}\right ) x^3\right )}{\sqrt [3]{-1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \left (-4+2 \left (3-2 \sqrt [3]{-1}\right ) x^3\right )}{\sqrt [3]{-1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{12 \sqrt [3]{-x^2+x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {x^3 \left (4-2 \left (3+2 \sqrt [3]{-1}\right ) x^3\right )}{\sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{12 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {(1-x) x}{18 \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1} \left (2-3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \left (2+3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}+\frac {(-1)^{2/3} \left (2-3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \left (2+3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}-\frac {(1-x) x^2}{2 \sqrt [3]{-x^2+x^3}}-\frac {x^3}{2 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2+16 x^3}{\left (-1-x^3\right ) \sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}}+\frac {\left (7 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{9 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (3-2 \sqrt [3]{-1}\right )+\left (13-i \sqrt {3}\right ) x^3}{\sqrt [3]{-1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (3+2 \sqrt [3]{-1}\right )+\left (19+5 i \sqrt {3}\right ) x^3}{\sqrt [3]{-1+x^3} \left (-1+\sqrt [3]{-1} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (3+2 (-1)^{2/3}\right )+\left (13+i \sqrt {3}\right ) x^3}{\sqrt [3]{-1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {-2 \left (3-2 (-1)^{2/3}\right )+\left (19-5 i \sqrt {3}\right ) x^3}{\sqrt [3]{-1+x^3} \left (-1-(-1)^{2/3} x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {5 (1-x) x}{6 \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1} \left (2-3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \left (2+3 \sqrt [3]{-1}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}+\frac {(-1)^{2/3} \left (2-3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \left (2+3 (-1)^{2/3}\right ) (1-x) x}{18 \sqrt [3]{-x^2+x^3}}-\frac {(1-x) x^2}{2 \sqrt [3]{-x^2+x^3}}-\frac {x^3}{2 \sqrt [3]{-x^2+x^3}}+\frac {\left (7 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{27 \sqrt [3]{-x^2+x^3}}-\frac {\left (4 \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{9 \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 )}{2 \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 )}{2 \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 )}{2 \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 )}{2 \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 )}{2 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \left (-17+7 i \sqrt {3}\right ) \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1} \left (-5+7 i \sqrt {3}\right ) \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \left (5+7 i \sqrt {3}\right ) \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}}+\frac {\left ((-1)^{2/3} \left (17+7 i \sqrt {3}\right ) \sqrt [3]{-1+x} x^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,\sqrt [3]{x}\right )}{36 \sqrt [3]{-x^2+x^3}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.09 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\frac {x^{2/3} \left (-12 \sqrt [3]{x}+2\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2\ 2^{2/3} \sqrt [3]{-1+x} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )+2^{2/3} \sqrt [3]{-1+x} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )+4 \sqrt [3]{-1+x} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]-4 \sqrt [3]{-1+x} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{24 \sqrt [3]{(-1+x) x^2}} \]
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Time = 0.72 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-3 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-12 x}{24 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\) | \(237\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.33 (sec) , antiderivative size = 1164, normalized size of antiderivative = 4.53 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\text {Too large to display} \]
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Not integrable
Time = 0.76 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.13 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\int \frac {x^{3}}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\int { \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 6.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx=\int \frac {x^3}{\left (x^6-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]
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