Optimal. Leaf size=214 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+1}-x\right )}{3\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3+1}+x}\right )}{2^{2/3} \sqrt {3}}+\frac {2 \left (x^3+1\right )^{2/3} \left (3 x^3-2\right )}{5 x^5}+\frac {1}{6} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+1} x+2^{2/3} \left (x^3+1\right )^{2/3}+x^2\right )}{6\ 2^{2/3}} \]
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Rubi [C] time = 0.39, antiderivative size = 103, normalized size of antiderivative = 0.48, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6725, 264, 277, 239, 429} \begin {gather*} \frac {5}{2} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{2}\right )+2 \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {4 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {4 \left (x^3+1\right )^{5/3}}{5 x^5}+\frac {2 \left (x^3+1\right )^{2/3}}{x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 264
Rule 277
Rule 429
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (8-4 x^3+x^6\right )}{x^6 \left (2+x^3\right )} \, dx &=\int \left (\frac {4 \left (1+x^3\right )^{2/3}}{x^6}-\frac {4 \left (1+x^3\right )^{2/3}}{x^3}+\frac {5 \left (1+x^3\right )^{2/3}}{2+x^3}\right ) \, dx\\ &=4 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx-4 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+5 \int \frac {\left (1+x^3\right )^{2/3}}{2+x^3} \, dx\\ &=\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {4 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {5}{2} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{2}\right )-4 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {4 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {5}{2} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{2}\right )-\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2 \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.23, size = 154, normalized size = 0.72 \begin {gather*} \frac {1}{8} \left (x^4 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-x^3,-\frac {x^3}{2}\right )+\frac {16 \left (x^3+1\right )^{2/3} \left (3 x^3-2\right )}{5 x^5}-2 \sqrt [3]{2} \left (-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{x^3+1}}\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 {2^{2/3} x}{\sqrt [3]{x^3+1}}+\frac {\sqrt [3]{2} x^2}{\left (x^3+1\right )^{2/3}}+2\right )\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.48, size = 214, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+1}-x\right )}{3\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3+1}+x}\right )}{2^{2/3} \sqrt {3}}+\frac {2 \left (x^3+1\right )^{2/3} \left (3 x^3-2\right )}{5 x^5}+\frac {1}{6} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3+1} x+2^{2/3} \left (x^3+1\right )^{2/3}+x^2\right )}{6\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 27.89, size = 361, normalized size = 1.69 \begin {gather*} \frac {100 \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 ) + 50 \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 ) - 25 \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 ) + 120 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - 60 \, x^{5} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 144 \, {\left (3 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{360 \, 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^{6} - 4 \, x^{3} + 8\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (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 [F] time = 1.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{6}-4 x^{3}+8\right )}{x^{6} \left (x^{3}+2\right )}\, dx \end {gather*}
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^{6} - 4 \, x^{3} + 8\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (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^6-4\,x^3+8\right )}{x^6\,\left (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^{2} + 2 x + 2\right ) \left (x^{4} - 2 x^{3} + 2 x^{2} - 4 x + 4\right )}{x^{6} \left (x^{3} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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