Optimal. Leaf size=143 \[ -\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1} x+\sqrt [3]{2} \left (x^3-1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3} \left (-2 x^6+2 x^3-5\right )}{10 x^8} \]
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Rubi [A] time = 0.42, antiderivative size = 277, normalized size of antiderivative = 1.94, number of steps used = 23, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {28, 586, 580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )}{12 \sqrt [3]{2}}+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}-\frac {\left (x^3-1\right )^{2/3}}{2 x^8}+\frac {11 \left (x^3-1\right )^{2/3}}{20 x^5}-\frac {79 \left (x^3-1\right )^{2/3}}{80 x^2}+\frac {1}{3} 2^{2/3} \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 )-\frac {\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 )}{24 \sqrt [3]{2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 12
Rule 28
Rule 31
Rule 200
Rule 204
Rule 377
Rule 580
Rule 583
Rule 586
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx &=\int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )^2}{x^9 \left (1+x^3\right )} \, dx\\ &=\frac {1}{8} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+x^3\right )} \, dx+2 \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9 \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}-\frac {\left (-1+x^3\right )^{2/3}}{20 x^5}+\frac {1}{40} \int \frac {9-x^3}{x^3 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {1}{4} \int \frac {12-4 x^3}{x^6 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}+\frac {9 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {1}{80} \int -\frac {20}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {1}{20} \int \frac {-44+36 x^3}{x^3 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {1}{40} \int \frac {160}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+4 \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{12} \operatorname {Subst}\left (\int \frac {2+\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+4 \operatorname {Subst}\left (\int \frac {1}{1-2 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {2+\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}+2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{4 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\left (2\ 2^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{12 \sqrt [3]{2}}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{24 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+\frac {2^{2/3} x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.27, size = 139, normalized size = 0.97 \begin {gather*} \frac {1}{30} \left (5\ 2^{2/3} \left (-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )\right )-\frac {3 \left (x^3-1\right )^{2/3} \left (2 x^6-2 x^3+5\right )}{x^8}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.37, size = 143, normalized size = 1.00 \begin {gather*} -\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-2 x\right )+\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\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 {\left (x^3-1\right )^{2/3} \left (-2 x^6+2 x^3-5\right )}{10 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.55, size = 276, normalized size = 1.93 \begin {gather*} -\frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{8} \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 ) - 10 \, \left (-4\right )^{\frac {1}{3}} x^{8} \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 ) + 5 \, \left (-4\right )^{\frac {1}{3}} x^{8} \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 ) + 9 \, {\left (2 \, x^{6} - 2 \, x^{3} + 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{8}} \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} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )} x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.62, size = 628, normalized size = 4.39
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^{6} + 4 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )} x^{9}}\,{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^6+4\,x^3+4\right )}{x^9\,\left (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 )^{2}}{x^{9} \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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