Optimal. Leaf size=254 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{6\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (8-13 x^3\right )}{20 x^5}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{12\ 2^{2/3}} \]
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Rubi [F] time = 0.74, 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 )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}-\frac {\left (-1+x^3\right )^{2/3}}{3 (1+x)}+\frac {(-2+x) \left (-1+x^3\right )^{2/3}}{3 \left (1-x+x^2\right )}+\frac {\left (-1+x^3\right )^{2/3}}{2 \left (-2+x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx\right )+\frac {1}{3} \int \frac {(-2+x) \left (-1+x^3\right )^{2/3}}{1-x+x^2} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {1}{3} \int \left (\frac {\left (1+i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {\left (-1+x^3\right )^{2/3} \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{2 \left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{4 \left (1-x^3\right )^{2/3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{-1-i \sqrt {3}+2 x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.50, size = 236, normalized size = 0.93 \begin {gather*} \frac {2 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{x^3-1}}\right )-8 \sqrt [3]{2} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )+8 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+2^{2/3}\right )+4 \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )}{12\ 2^{2/3}}+\left (x^3-1\right )^{2/3} \left (\frac {2}{5 x^5}-\frac {13}{20 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 254, normalized size = 1.00 \begin {gather*} \frac {\left (8-13 x^3\right ) \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.31, size = 521, normalized size = 2.05 \begin {gather*} -\frac {80 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) + 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 80 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 40 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \, {\left (13 \, x^{3} - 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{720 \, x^{5}} \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*} \int \frac {{\left (x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 42.03, size = 1854, normalized size = 7.30
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1854\) |
trager | \(\text {Expression too large to display}\) | \(1899\) |
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 {{\left (x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 2\right )} x^{6}}\,{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 {{\left (x^3-1\right )}^{2/3}\,\left (x^3+4\right )}{x^6\,\left (-x^6+x^3+2\right )} \,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 {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 4\right )}{x^{6} \left (x + 1\right ) \left (x^{3} - 2\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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