Optimal. Leaf size=149 \[ -\frac {2}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{3 x^3+1}-2 x\right )+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{3 x^3+1}+x}\right )}{\sqrt {3}}+\frac {\left (3 x^3+1\right )^{2/3} \left (1-2 x^3\right )}{5 x^5}+\frac {1}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{3 x^3+1} x+\sqrt [3]{2} \left (3 x^3+1\right )^{2/3}+2 x^2\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 158, normalized size of antiderivative = 1.06, 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*} -\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}\right )+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}-\frac {2 \left (3 x^3+1\right )^{2/3}}{5 x^2}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}+\frac {2^{2/3} x^2}{\left (3 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 (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {4+24 x^3}{x^3 \left (1+x^3\right ) \sqrt [3]{1+3 x^3}} \, dx\\ &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}-\frac {1}{10} \int -\frac {40}{\left (1+x^3\right ) \sqrt [3]{1+3 x^3}} \, dx\\ &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}+4 \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{1+3 x^3}} \, dx\\ &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}+4 \operatorname {Subst}\left (\int \frac {1}{1-2 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+3 x^3}}\right )\\ &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{1+3 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+3 x^3}}\right )\\ &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{1+3 x^3}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+3 x^3}}\right )+\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+3 x^3}}\right )\\ &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{1+3 x^3}}\right )+\frac {1}{3} 2^{2/3} \log \left (1+\frac {2^{2/3} x^2}{\left (1+3 x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{1+3 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+3 x^3}}\right )\\ &=\frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{1+3 x^3}}\right )+\frac {1}{3} 2^{2/3} \log \left (1+\frac {2^{2/3} x^2}{\left (1+3 x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{1+3 x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.27, size = 130, normalized size = 0.87 \begin {gather*} \frac {1}{3} 2^{2/3} \left (-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+3}}+1}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+3}}+\frac {2^{2/3} x^2}{\left (x^3+3\right )^{2/3}}+1\right )\right )+\left (3 x^3+1\right )^{2/3} \left (\frac {1}{5 x^5}-\frac {2}{5 x^2}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.38, size = 149, normalized size = 1.00 \begin {gather*} \frac {\left (1-2 x^3\right ) \left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+3 x^3}}\right )}{\sqrt {3}}-\frac {2}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+3 x^3}\right )+\frac {1}{3} 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+3 x^3}+\sqrt [3]{2} \left (1+3 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.92, size = 279, normalized size = 1.87 \begin {gather*} \frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{6} + 39 \, x^{3} + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{6} - 3 \, x^{3} - 1\right )}}\right ) + 10 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, 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^{5} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{3} + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 9 \, {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{3} - 1\right )}}{45 \, 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 (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (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 = 18.03, size = 643, normalized size = 4.32
method | result | size |
risch | \(-\frac {6 x^{6}-x^{3}-1}{5 x^{5} \left (3 x^{3}+1\right )^{\frac {1}{3}}}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+24 \left (3 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +8 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+42 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{2}+15 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-2 \left (3 x^{3}+1\right )^{\frac {2}{3}} x +5 \RootOf \left (\textit {\_Z}^{3}+4\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\frac {2 \RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+45 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+12 \left (3 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +4 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+3 \left (3 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{2}-5 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-75 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-7 \left (3 x^{3}+1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{3}+4\right )-15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+36 \textit {\_Z}^{2}\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}\) | \(643\) |
trager | \(\text {Expression too large to display}\) | \(1164\) |
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 (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (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 )\,{\left (3\,x^3+1\right )}^{2/3}}{x^6\,\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 (x - 1\right ) \left (3 x^{3} + 1\right )^{\frac {2}{3}} \left (x^{2} + x + 1\right )}{x^{6} \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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