Optimal. Leaf size=257 \[ -\frac {\left (-x^2+x^3\right )^{2/3}}{2 (-1+x) x}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {1}{6} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]-\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 2.12, antiderivative size = 1933, normalized size of antiderivative = 7.52, number of steps
used = 42, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2081, 6857,
21, 49, 52, 61, 129, 490, 596, 544, 245, 384} \begin {gather*} -\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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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} \text {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}} \end {gather*}
Warning: Unable to verify antiderivative.
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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*} \int \frac {x^3}{\sqrt [3]{-x^2+x^3} \left (-1+x^6\right )} \, dx &=\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^{4/3}}{\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^{4/3}}{(-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)^{4/3}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {x^{4/3}}{\sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{6 \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 ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{9 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{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 {\sqrt [3]{x}}{(-1-x) \sqrt [3]{-1+x}} \, dx}{6 \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}}-2 \frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\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 ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}+2 \frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}} \, dx}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (-1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-1+x} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {(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 ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{27 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{18 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{6 \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^{2/3}} \, dx}{6 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \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 {1}{\sqrt [3]{-1+x} x^{2/3} \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 {1}{\sqrt [3]{-1+x} x^{2/3} \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 {1}{\sqrt [3]{-1+x} x^{2/3} \left (1-(-1)^{2/3} x\right )} \, dx}{6 \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}}-2 \left (-\frac {\sqrt [3]{-1} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{18 \sqrt [3]{-x^2+x^3}}\right )+2 \left (-\frac {(-1)^{2/3} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{18 \sqrt [3]{-x^2+x^3}}\right )\\ &=-\frac {5 (1-x) x}{6 \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 {4 \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{9 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {2 \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{9 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{12 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {2 \sqrt [3]{-1+x} x^{2/3} \log (x)}{27 \sqrt [3]{-x^2+x^3}}-2 \left (-\frac {\sqrt [3]{-1} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+2 \left (-\frac {(-1)^{2/3} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1-(-1)^{2/3}} \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 {5 (1-x) x}{6 \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 {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{3 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{6 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{12 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{18 \sqrt [3]{-x^2+x^3}}-2 \left (-\frac {\sqrt [3]{-1} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+2 \left (-\frac {(-1)^{2/3} (1-x) x}{6 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{6 \sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{12 \sqrt [3]{-x^2+x^3}}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3} \log (x)}{36 \sqrt [3]{-x^2+x^3}}\right )+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1+\sqrt [3]{-1} x\right )}{12 \sqrt [3]{1+\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (1-(-1)^{2/3} x\right )}{12 \sqrt [3]{1-(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 281, normalized size = 1.09 \begin {gather*} \frac {x^{2/3} \left (-12 \sqrt [3]{x}+2\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} \text {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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 13.62, size = 4486, normalized size = 17.46
method | result | size |
trager | \(\text {Expression too large to display}\) | \(4486\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.61, size = 2380, normalized size = 9.26 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{\left (x^6-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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