Optimal. Leaf size=78 \[ \frac {\left (x^3-1\right )^{2/3} \left (7 x^3-2\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^4-\text {$\#$1}}\& \right ] \]
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Rubi [C] time = 0.14, antiderivative size = 223, normalized size of antiderivative = 2.86, number of steps used = 10, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {28, 1520, 277, 239, 1428, 430, 429} \begin {gather*} -\frac {2 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},1;\frac {4}{3};x^3,\frac {2 x^3}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {2 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},1;\frac {4}{3};x^3,\frac {2 x^3}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 28
Rule 239
Rule 277
Rule 429
Rule 430
Rule 1428
Rule 1520
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (1-x^3+x^6\right )} \, dx &=\int \frac {\left (-1+x^3\right )^{8/3}}{x^6 \left (1-x^3+x^6\right )} \, dx\\ &=-\int \frac {\left (-1+x^3\right )^{5/3}}{x^6} \, dx+\int \frac {\left (-1+x^3\right )^{5/3}}{1-x^3+x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {(2 i) \int \frac {\left (-1+x^3\right )^{5/3}}{-1-i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}+\frac {(2 i) \int \frac {\left (-1+x^3\right )^{5/3}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}-\int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (2 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{5/3}}{-1-i \sqrt {3}+2 x^3} \, dx}{\sqrt {3} \left (1-x^3\right )^{2/3}}-\frac {\left (2 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{5/3}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3} \left (1-x^3\right )^{2/3}}-\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},1;\frac {4}{3};x^3,\frac {2 x^3}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (i+\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}+\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},1;\frac {4}{3};x^3,\frac {2 x^3}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 1.94, size = 415, normalized size = 5.32 \begin {gather*} \left (x^3-1\right )^{2/3} \left (\frac {7}{10 x^2}-\frac {1}{5 x^5}\right )+\frac {i \left (-\frac {2 \log \left (\sqrt [3]{\sqrt {3}+i}-\frac {\sqrt [3]{\sqrt {3}-i} x}{\sqrt [3]{x^3-1}}\right )}{\sqrt [3]{\frac {\sqrt {3}-i}{\sqrt {3}+i}}}+2 \sqrt [3]{\frac {\sqrt {3}-i}{\sqrt {3}+i}} \log \left (\sqrt [3]{\sqrt {3}-i}-\frac {\sqrt [3]{\sqrt {3}+i} x}{\sqrt [3]{x^3-1}}\right )+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt {3}-i\right )^{2/3} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+\frac {\left (\sqrt {3}-i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {3}+i\right )^{2/3}\right )}{\left (\frac {1}{2} \left (\sqrt {3}-i\right )\right )^{2/3}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt {3}+i\right )^{2/3} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+\frac {\left (\sqrt {3}+i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {3}-i\right )^{2/3}\right )}{\left (\frac {1}{2} \left (\sqrt {3}+i\right )\right )^{2/3}}\right )}{6 \sqrt {3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.21, size = 78, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 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 [B] time = 26.34, size = 1461, normalized size = 18.73
method | result | size |
risch | \(\frac {7 x^{6}-9 x^{3}+2}{10 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} \ln \left (\frac {-177147 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{7} x^{3}-2916 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x^{3}-243 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x^{2}+27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {2}{3}} x +729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4}-12 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}-2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+6 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x +1\right )}\right )+1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} \ln \left (\frac {729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+81 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x -18 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x^{3}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+x \left (x^{3}-1\right )^{\frac {2}{3}}+9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2}}{\left (27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x +1\right )}\right )-243 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} \ln \left (-\frac {27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x -1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x +1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +1\right )}\right )+9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \ln \left (\frac {729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+81 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x -18 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x^{3}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+x \left (x^{3}-1\right )^{\frac {2}{3}}+9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2}}{\left (27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x -1\right ) \left (6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} x +1\right )}\right )+\RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \ln \left (-\frac {177147 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{7} x^{3}+6561 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{5} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x^{3}+243 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{3} x +54 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x -1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x +1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +1\right )}\right )-2 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) \ln \left (-\frac {27 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+3 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )}{\left (1458 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x -1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right ) x +1\right ) \left (729 \RootOf \left (19683 \textit {\_Z}^{6}+243 \textit {\_Z}^{3}+1\right )^{4} x +1\right )}\right )\) | \(1461\) |
trager | \(\text {Expression too large to display}\) | \(4455\) |
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} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 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^6-2\,x^3+1\right )}{x^6\,\left (x^6-x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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