Optimal. Leaf size=257 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-3 \text {$\#$1}^3+3\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {\left (x^3-x^2\right )^{2/3}}{2 (x-1) x}-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{12 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{2 \sqrt [3]{2} \sqrt {3}} \]
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Rubi [B] time = 2.32, antiderivative size = 1568, normalized size of antiderivative = 6.10, number of steps used = 62, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2056, 6725, 21, 47, 50, 59, 105, 91}
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Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 21
Rule 47
Rule 50
Rule 59
Rule 91
Rule 105
Rule 2056
Rule 6725
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 [C] time = 4.74, size = 928, normalized size = 3.61 \begin {gather*} \frac {x \left (\frac {2^{2/3} \left (6 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1}{\sqrt {3}}\right )+6 \sqrt {3} \sqrt [3]{1-i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{1-i \sqrt {3}}}+1}{\sqrt {3}}\right )-6 \sqrt [6]{3} \sqrt [3]{3-i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{3-i \sqrt {3}}}+1}{\sqrt {3}}\right )+6 \sqrt {3} \sqrt [3]{1+i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{1+i \sqrt {3}}}+1}{\sqrt {3}}\right )-6 \sqrt [6]{3} \sqrt [3]{3+i \sqrt {3}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}}{\sqrt [3]{3+i \sqrt {3}}}+1}{\sqrt {3}}\right )-6 \log \left (1-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )-6 \sqrt [3]{1-i \sqrt {3}} \log \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )-6 \sqrt [3]{1+i \sqrt {3}} \log \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )+2\ 3^{2/3} \sqrt [3]{3-i \sqrt {3}} \log \left (\sqrt [3]{3-i \sqrt {3}}-\sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}\right )+2\ 3^{2/3} \sqrt [3]{3+i \sqrt {3}} \log \left (\sqrt [3]{3+i \sqrt {3}}-\sqrt [3]{6} \sqrt [3]{\frac {x}{x-1}}\right )+3 \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1\right )+3 \sqrt [3]{1-i \sqrt {3}} \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2-2 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (1-i \sqrt {3}\right )^{2/3}\right )+3 \sqrt [3]{1+i \sqrt {3}} \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2+2 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (1+i \sqrt {3}\right )^{2/3}\right )-3^{2/3} \sqrt [3]{3-i \sqrt {3}} \log \left (6^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{18-6 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (3-i \sqrt {3}\right )^{2/3}\right )-3^{2/3} \sqrt [3]{3+i \sqrt {3}} \log \left (6^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{18+6 i \sqrt {3}} \sqrt [3]{\frac {x}{x-1}}+\left (3+i \sqrt {3}\right )^{2/3}\right )\right )}{\sqrt [3]{\frac {x}{x-1}}}-36\right )}{72 \sqrt [3]{(x-1) x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.61, size = 257, normalized size = 1.00 \begin {gather*} -\frac {\left (-x^2+x^3\right )^{2/3}}{2 (-1+x) x}+\frac {\tan ^{-1}\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 ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 2380, normalized size = 9.26
result too large to display
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*} \int \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 141.39, size = 9712, normalized size = 37.79
method | result | size |
trager | \(\text {Expression too large to display}\) | \(9712\) |
risch | \(\text {Expression too large to display}\) | \(15043\) |
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*} \int \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \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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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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