Optimal. Leaf size=145 \[ \frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{12\ 2^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{24\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^6+8 x^3+10\right )}{40 x^8} \]
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Rubi [C] time = 0.35, antiderivative size = 143, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6725, 271, 264, 277, 239, 430, 429} \begin {gather*} -\frac {5 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{8 \left (1-x^3\right )^{2/3}}+\frac {5}{8} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {5 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\left (x^3-1\right )^{5/3}}{4 x^8}-\frac {9 \left (x^3-1\right )^{5/3}}{20 x^5}+\frac {5 \left (x^3-1\right )^{2/3}}{8 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 264
Rule 271
Rule 277
Rule 429
Rule 430
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3+x^6\right )}{x^9 \left (-2+x^3\right )} \, dx &=\int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^9}-\frac {3 \left (-1+x^3\right )^{2/3}}{2 x^6}-\frac {5 \left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {5 \left (-1+x^3\right )^{2/3}}{4 \left (-2+x^3\right )}\right ) \, dx\\ &=-\left (\frac {5}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {5}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-\frac {3}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx\\ &=\frac {5 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}-\frac {3 \left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {3}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-\frac {5}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {\left (5 \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{4 \left (1-x^3\right )^{2/3}}\\ &=\frac {5 \left (-1+x^3\right )^{2/3}}{8 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{4 x^8}-\frac {9 \left (-1+x^3\right )^{5/3}}{20 x^5}-\frac {5 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{8 \left (1-x^3\right )^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {5}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 138, normalized size = 0.95 \begin {gather*} \frac {\left (x^3-1\right )^{2/3} \left (7 x^6+8 x^3+10\right )}{40 x^8}-\frac {5 \left (-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} x^2}{\left (1-x^3\right )^{2/3}}+2\right )\right )}{24\ 2^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.34, size = 145, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (10+8 x^3+7 x^6\right )}{40 x^8}-\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 = 3.36, size = 277, normalized size = 1.91 \begin {gather*} \frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{8} \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^{8} \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^{8} \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 ) + 36 \, {\left (7 \, x^{6} + 8 \, x^{3} + 10\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, 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} + x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 16.63, size = 744, normalized size = 5.13
method | result | size |
risch | \(\frac {7 x^{9}+x^{6}+2 x^{3}-10}{40 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}-\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}-3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-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 ) x^{3}-6 x \left (x^{3}-1\right )^{\frac {2}{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 )}{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}-3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-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 ) x^{3}-6 x \left (x^{3}-1\right )^{\frac {2}{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 )}{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}+\frac {5 \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 ) \ln \left (-\frac {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 ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+36 \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}+30 \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 +4 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+30 \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}+9 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+36 \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}+8 x \left (x^{3}-1\right )^{\frac {2}{3}}-6 \RootOf \left (\textit {\_Z}^{3}-2\right )-24 \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}-2}\right )}{4}\) | \(744\) |
trager | \(\text {Expression too large to display}\) | \(1120\) |
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} + x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\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+x^3+4\right )}{x^9\,\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^{6} + x^{3} + 4\right )}{x^{9} \left (x^{3} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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