Optimal. Leaf size=131 \[ \frac {\sqrt [3]{1+x^6}}{x}+\frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^6}-\sqrt [3]{2} \left (1+x^6\right )^{2/3}\right )}{3\ 2^{2/3}} \]
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Rubi [F]
time = 0.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx &=\int \left (-\frac {\sqrt [3]{1+x^6}}{x^2}-\frac {2 \sqrt [3]{1+x^6}}{3 (1+x)}+\frac {2 (1+x) \sqrt [3]{1+x^6}}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=-\left (\frac {2}{3} \int \frac {\sqrt [3]{1+x^6}}{1+x} \, dx\right )+\frac {2}{3} \int \frac {(1+x) \sqrt [3]{1+x^6}}{1-x+x^2} \, dx-\int \frac {\sqrt [3]{1+x^6}}{x^2} \, dx\\ &=\frac {\, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-x^6\right )}{x}-\frac {2}{3} \int \frac {\sqrt [3]{1+x^6}}{1+x} \, dx+\frac {2}{3} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+x^6}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+x^6}}{-1+i \sqrt {3}+2 x}\right ) \, dx\\ &=\frac {\, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-x^6\right )}{x}-\frac {2}{3} \int \frac {\sqrt [3]{1+x^6}}{1+x} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt [3]{1+x^6}}{-1-i \sqrt {3}+2 x} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt [3]{1+x^6}}{-1+i \sqrt {3}+2 x} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 131, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{1+x^6}}{x}+\frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^6}-\sqrt [3]{2} \left (1+x^6\right )^{2/3}\right )}{3\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-1\right ) \left (x^{6}+1\right )^{\frac {1}{3}}}{x^{2} \left (x^{3}+1\right )}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs.
\(2 (100) = 200\).
time = 15.50, size = 327, normalized size = 2.50 \begin {gather*} \frac {2 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} \left (-2\right )^{\frac {2}{3}} {\left (x^{14} - 14 \, x^{11} + 6 \, x^{8} - 14 \, x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 6 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} {\left (x^{13} - 2 \, x^{10} - 6 \, x^{7} - 2 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{18} - 30 \, x^{15} + 51 \, x^{12} - 52 \, x^{9} + 51 \, x^{6} - 30 \, x^{3} + 1\right )}}{3 \, {\left (x^{18} + 6 \, x^{15} - 93 \, x^{12} + 20 \, x^{9} - 93 \, x^{6} + 6 \, x^{3} + 1\right )}}\right ) + 2 \, \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {6 \, \left (-2\right )^{\frac {1}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} - \left (-2\right )^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )} - 6 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x^{3} + 1}\right ) - \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {3 \, \left (-2\right )^{\frac {2}{3}} {\left (x^{7} - 4 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \left (-2\right )^{\frac {1}{3}} {\left (x^{12} - 14 \, x^{9} + 6 \, x^{6} - 14 \, x^{3} + 1\right )} - 12 \, {\left (x^{8} - x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 18 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{18 \, 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 {\sqrt [3]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{2} \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 [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 = -0.01 \begin {gather*} \int \frac {\left (x^3-1\right )\,{\left (x^6+1\right )}^{1/3}}{x^2\,\left (x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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