Optimal. Leaf size=211 \[ \frac {3 \left (5+6 x+9 x^2\right ) \left (-x^2+x^3\right )^{2/3}}{40 x^4}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.75, antiderivative size = 769, normalized size of antiderivative = 3.64, number of steps
used = 21, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2081, 6857,
129, 491, 597, 12, 384} \begin {gather*} -\frac {\sqrt [3]{x-1} x^{2/3} \text {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} \text {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} \text {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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 129
Rule 384
Rule 491
Rule 597
Rule 2081
Rule 6857
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx &=\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 {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {2}{3}-2 x}{(-1-x) \sqrt [3]{-1+x} x^{8/3}} \, dx}{8 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {-\frac {2}{3} \left (3+4 \sqrt [3]{-1}\right )+2 \sqrt [3]{-1} x}{\sqrt [3]{-1+x} x^{8/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{8 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {-\frac {2}{3} \left (3-4 (-1)^{2/3}\right )-2 (-1)^{2/3} x}{\sqrt [3]{-1+x} x^{8/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{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 (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {34}{9}-\frac {2 x}{3}}{(-1-x) \sqrt [3]{-1+x} x^{5/3}} \, dx}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {2}{9} \left (5+16 i \sqrt {3}\right )-\frac {2}{3} \sqrt [3]{-1} \left (3+4 \sqrt [3]{-1}\right ) x}{\sqrt [3]{-1+x} x^{5/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {2}{9} \left (5-16 i \sqrt {3}\right )+\frac {2}{3} (-1)^{2/3} \left (3-4 (-1)^{2/3}\right ) x}{\sqrt [3]{-1+x} x^{5/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{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 (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}+\frac {\left (9 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {80}{27 (-1-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{80 \sqrt [3]{-x^2+x^3}}+\frac {\left (9 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {80}{27 \sqrt [3]{-1+x} x^{2/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{80 \sqrt [3]{-x^2+x^3}}+\frac {\left (9 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {80}{27 \sqrt [3]{-1+x} x^{2/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{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 (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \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}{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^{2/3} \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^{2/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \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 (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \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 )}{\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 )}{\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 )}{\sqrt {3} \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]{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 (\frac {\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 (\frac {\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 (-1-x)}{6 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{6 \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 )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 251, normalized size = 1.19 \begin {gather*} \frac {-45-9 x-27 x^2+81 x^3-20\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} x^{8/3} \text {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}} \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 = 37.93, size = 3474, normalized size = 16.46
method | result | size |
trager | \(\text {Expression too large to display}\) | \(3474\) |
risch | \(\text {Expression too large to display}\) | \(4024\) |
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 = 0.42, size = 895, normalized size = 4.24 \begin {gather*} \frac {40 \, x^{4} \cos \left (\frac {1}{9} \, \pi \right ) \log \left (\frac {16 \, {\left (x^{2} - {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}\right )}}{x^{2}}\right ) - 160 \, x^{4} \arctan \left (\frac {8 \, {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{3} - x \cos \left (\frac {1}{9} \, \pi \right )\right )} \sin \left (\frac {1}{9} \, \pi \right ) + \sqrt {3} x + 2 \, {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3} x\right )} \sqrt {\frac {x^{2} - {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) + 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}} - 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} + 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )}}{16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{4} - 16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + 3 \, x}\right ) \sin \left (\frac {1}{9} \, \pi \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 ) + 80 \, {\left (\sqrt {3} x^{4} \cos \left (\frac {1}{9} \, \pi \right ) + x^{4} \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (\frac {8 \, {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{3} - x \cos \left (\frac {1}{9} \, \pi \right )\right )} \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3} x - 2 \, {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right )^{2} - 2 \, x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3} x\right )} \sqrt {\frac {x^{2} + {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + x\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}} + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (2 \, \sqrt {3} \cos \left (\frac {1}{9} \, \pi \right )^{2} - 2 \, \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - \sqrt {3}\right )}}{16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{4} - 16 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + 3 \, x}\right ) - 80 \, {\left (\sqrt {3} x^{4} \cos \left (\frac {1}{9} \, \pi \right ) - x^{4} \sin \left (\frac {1}{9} \, \pi \right )\right )} \arctan \left (-\frac {2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x \sqrt {\frac {x^{2} + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}} - x + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{2 \, x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right )}\right ) + 20 \, {\left (\sqrt {3} x^{4} \sin \left (\frac {1}{9} \, \pi \right ) - x^{4} \cos \left (\frac {1}{9} \, \pi \right )\right )} \log \left (\frac {64 \, {\left (x^{2} + {\left (2 \, \sqrt {3} x \cos \left (\frac {1}{9} \, \pi \right ) \sin \left (\frac {1}{9} \, \pi \right ) - 2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} + x\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}\right )}}{x^{2}}\right ) - 20 \, {\left (\sqrt {3} x^{4} \sin \left (\frac {1}{9} \, \pi \right ) + x^{4} \cos \left (\frac {1}{9} \, \pi \right )\right )} \log \left (\frac {64 \, {\left (x^{2} + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (2 \, x \cos \left (\frac {1}{9} \, \pi \right )^{2} - x\right )} + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}\right )}}{x^{2}}\right ) + 9 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}} {\left (9 \, x^{2} + 6 \, x + 5\right )}}{120 \, x^{4}} \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 {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right )} \left (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 [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 33.13, size = 1007, normalized size = 4.77 \begin {gather*} \text {Too large to display} \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 {1}{x^3\,\left (x^3+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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