Optimal. Leaf size=134 \[ -3^{2/3} \log \left (3^{2/3} \sqrt [3]{x^3+1}-3 x\right )+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (-19 x^3-4\right )}{10 x^5}+\frac {1}{2} 3^{2/3} \log \left (3^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 137, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}-\frac {19 \left (x^3+1\right )^{2/3}}{10 x^2}+\frac {1}{2} 3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 200
Rule 204
Rule 377
Rule 580
Rule 583
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {19+7 x^3}{x^3 \sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int \frac {90}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-9 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-9 \operatorname {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3 \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-3 \operatorname {Subst}\left (\int \frac {-2-\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )-\left (3\ 3^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 124, normalized size = 0.93 \begin {gather*} 3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {\left (x^3+1\right )^{2/3} \left (19 x^3+4\right )}{10 x^5}+\frac {1}{2} 3^{2/3} \left (\log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )-2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 134, normalized size = 1.00 \begin {gather*} -3^{2/3} \log \left (3^{2/3} \sqrt [3]{x^3+1}-3 x\right )+3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (-19 x^3-4\right )}{10 x^5}+\frac {1}{2} 3^{2/3} \log \left (3^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.27, size = 274, normalized size = 2.04 \begin {gather*} -\frac {10 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2 \, \sqrt {3} \left (-9\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 10 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-9\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-9\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )}}{2 \, x^{3} - 1}\right ) + 5 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {9 \, \left (-9\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-9\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 27 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) + 3 \, {\left (19 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, 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} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.25, size = 612, normalized size = 4.57
result too large to display
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} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\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 = -0.01 \begin {gather*} \int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3-2\right )}{x^6\,\left (2\,x^3-1\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} - 2\right )}{x^{6} \left (2 x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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