Optimal. Leaf size=150 \[ \frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{1-x^3}-x\right )}{12\ 2^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{1-x^3}+x}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} \left (29 x^3-4\right )}{40 x^5}-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{1-x^3} x+2^{2/3} \left (1-x^3\right )^{2/3}+x^2\right )}{24\ 2^{2/3}} \]
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Rubi [A] time = 0.16, antiderivative size = 157, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {5 \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+2^{2/3}\right )}{24\ 2^{2/3}} \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 )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}-\frac {1}{10} \int \frac {-29+31 x^3}{x^3 \sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {1}{40} \int -\frac {50}{\sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5}{4} \int \frac {1}{\sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{-2+x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{2}+x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}+\frac {5 \operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{2}-x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2 x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{24\ 2^{2/3}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{8 \sqrt [3]{2}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (2^{2/3}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{24\ 2^{2/3}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )}{4\ 2^{2/3}}\\ &=-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {5 \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{1-x^3}}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (2^{2/3}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{24\ 2^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 129, normalized size = 0.86 \begin {gather*} \left (\frac {29}{40 x^2}-\frac {1}{10 x^5}\right ) \left (1-x^3\right )^{2/3}-\frac {5 \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 )}{24\ 2^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.35, size = 150, normalized size = 1.00 \begin {gather*} \frac {\left (1-x^3\right )^{2/3} \left (-4+29 x^3\right )}{40 x^5}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1-x^3}}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {5 \log \left (-x+\sqrt [3]{2} \sqrt [3]{1-x^3}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1-x^3}+2^{2/3} \left (1-x^3\right )^{2/3}\right )}{24\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 3.21, size = 278, normalized size = 1.85 \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 (3 \, x^{4} - 2 \, x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{9} - 72 \, x^{6} + 36 \, x^{3} + 8\right )} + 12 \, \sqrt {3} {\left (9 \, x^{8} - 6 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (27 \, x^{9} - 36 \, 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 (3 \, x^{3} - 2\right )} - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{3 \, x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (9 \, x^{6} - 6 \, x^{3} - 4\right )} - 6 \, {\left (3 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{6} - 12 \, x^{3} + 4}\right ) + 36 \, {\left (29 \, x^{3} - 4\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{1440 \, 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 (4 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, 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 = 18.04, size = 791, normalized size = 5.27
method | result | size |
risch | \(-\frac {29 x^{6}-33 x^{3}+4}{40 x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}}+\frac {5 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+9 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -9 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+3 \left (-x^{3}+1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-2\right )-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{3 x^{3}-2}\right )}{24}-\frac {5 \ln \left (-\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-18 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{3 x^{3}-2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{24}-\frac {5 \ln \left (-\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-18 \left (-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{3 x^{3}-2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{4}\) | \(791\) |
trager | \(\text {Expression too large to display}\) | \(1147\) |
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 (4 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, 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 = -0.01 \begin {gather*} \int \frac {{\left (1-x^3\right )}^{2/3}\,\left (4\,x^3-1\right )}{x^6\,\left (3\,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 (4 x^{3} - 1\right )}{x^{6} \left (3 x^{3} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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