Optimal. Leaf size=102 \[ \log \left (\sqrt [3]{x^6+1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+1}+x}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^6+1} x+\left (x^6+1\right )^{2/3}+x^2\right )+\frac {\left (x^6+1\right )^{2/3} \left (2 x^6+15 x^3+2\right )}{10 x^5} \]
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Rubi [C] time = 1.29, antiderivative size = 288, normalized size of antiderivative = 2.82, number of steps used = 18, number of rules used = 8, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {6728, 245, 364, 275, 1438, 429, 465, 510} \begin {gather*} \frac {3 \left (-\sqrt {3}+i\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{\sqrt {3}+i}+\frac {3 \left (\sqrt {3}+i\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{-\sqrt {3}+i}+\frac {3 x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {2 x^6}{1-i \sqrt {3}}\right )}{2 \left (1-i \sqrt {3}\right )}+\frac {3 x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {2 x^6}{1+i \sqrt {3}}\right )}{2 \left (1+i \sqrt {3}\right )}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )+\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 245
Rule 275
Rule 364
Rule 429
Rule 465
Rule 510
Rule 1438
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx &=\int \left (\left (1+x^6\right )^{2/3}-\frac {\left (1+x^6\right )^{2/3}}{x^6}-\frac {3 \left (1+x^6\right )^{2/3}}{x^3}+\frac {3 \left (-1+2 x^3\right ) \left (1+x^6\right )^{2/3}}{1-x^3+x^6}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^6\right )^{2/3}}{x^3} \, dx\right )+3 \int \frac {\left (-1+2 x^3\right ) \left (1+x^6\right )^{2/3}}{1-x^3+x^6} \, dx+\int \left (1+x^6\right )^{2/3} \, dx-\int \frac {\left (1+x^6\right )^{2/3}}{x^6} \, dx\\ &=\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )+3 \int \left (\frac {2 \left (1+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^3}+\frac {2 \left (1+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^3}\right ) \, dx\\ &=\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )+6 \int \frac {\left (1+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^3} \, dx+6 \int \frac {\left (1+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^3} \, dx\\ &=\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )+6 \int \left (\frac {\left (i-\sqrt {3}\right ) \left (1+x^6\right )^{2/3}}{2 \left (i+\sqrt {3}+2 i x^6\right )}+\frac {x^3 \left (1+x^6\right )^{2/3}}{1-i \sqrt {3}+2 x^6}\right ) \, dx+6 \int \left (\frac {\left (-i-\sqrt {3}\right ) \left (1+x^6\right )^{2/3}}{2 \left (-i+\sqrt {3}-2 i x^6\right )}+\frac {x^3 \left (1+x^6\right )^{2/3}}{1+i \sqrt {3}+2 x^6}\right ) \, dx\\ &=\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )+6 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{1-i \sqrt {3}+2 x^6} \, dx+6 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{1+i \sqrt {3}+2 x^6} \, dx+\left (3 \left (i-\sqrt {3}\right )\right ) \int \frac {\left (1+x^6\right )^{2/3}}{i+\sqrt {3}+2 i x^6} \, dx-\left (3 \left (i+\sqrt {3}\right )\right ) \int \frac {\left (1+x^6\right )^{2/3}}{-i+\sqrt {3}-2 i x^6} \, dx\\ &=\frac {3 \left (i-\sqrt {3}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{i+\sqrt {3}}+\frac {3 \left (i+\sqrt {3}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{i-\sqrt {3}}+\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )+3 \operatorname {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x^3} \, dx,x,x^2\right )+3 \operatorname {Subst}\left (\int \frac {x \left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x^3} \, dx,x,x^2\right )\\ &=\frac {3 \left (i-\sqrt {3}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {2 x^6}{1-i \sqrt {3}},-x^6\right )}{i+\sqrt {3}}+\frac {3 \left (i+\sqrt {3}\right ) x F_1\left (\frac {1}{6};1,-\frac {2}{3};\frac {7}{6};-\frac {2 x^6}{1+i \sqrt {3}},-x^6\right )}{i-\sqrt {3}}+\frac {3 x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {2 x^6}{1-i \sqrt {3}}\right )}{2 \left (1-i \sqrt {3}\right )}+\frac {3 x^4 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};-x^6,-\frac {2 x^6}{1+i \sqrt {3}}\right )}{2 \left (1+i \sqrt {3}\right )}+\frac {\, _2F_1\left (-\frac {5}{6},-\frac {2}{3};\frac {1}{6};-x^6\right )}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+x \, _2F_1\left (-\frac {2}{3},\frac {1}{6};\frac {7}{6};-x^6\right )\\ \end {align*}
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Mathematica [F] time = 1.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^3 \left (1+x^6\right )^{2/3}}{x^6 \left (1-x^3+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.27, size = 102, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^6\right )^{2/3} \left (2+15 x^3+2 x^6\right )}{10 x^5}-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^6}}\right )+\log \left (-x+\sqrt [3]{1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 7.55, size = 147, normalized size = 1.44 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {1078 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (32 \, x^{6} + 605 \, x^{3} + 32\right )}}{8 \, x^{6} - 1331 \, x^{3} + 8}\right ) - 5 \, x^{5} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{3} + 1}\right ) - {\left (2 \, x^{6} + 15 \, x^{3} + 2\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{10 \, 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 (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{3} {\left (x^{3} - 1\right )}}{{\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 [C] time = 6.96, size = 416, normalized size = 4.08
method | result | size |
trager | \(\frac {\left (x^{6}+1\right )^{\frac {2}{3}} \left (2 x^{6}+15 x^{3}+2\right )}{10 x^{5}}+\ln \left (-\frac {6624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}-2763 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-1102 x^{6}-13248 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+4428 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x +6057 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{2}-9930 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3495 x \left (x^{6}+1\right )^{\frac {2}{3}}-1476 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-1653 x^{3}+6624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-2763 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1102}{x^{6}-x^{3}+1}\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {4959 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{6}+7167 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+2208 x^{6}-9918 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+4428 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x -10485 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{2}-2763 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2019 x \left (x^{6}+1\right )^{\frac {2}{3}}-1476 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}+736 x^{3}+4959 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7167 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2208}{x^{6}-x^{3}+1}\right )\) | \(416\) |
risch | \(\frac {2 x^{12}+15 x^{9}+4 x^{6}+15 x^{3}+2}{10 x^{5} \left (x^{6}+1\right )^{\frac {1}{3}}}-3 \ln \left (-\frac {-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-2 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{x^{6}-x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\ln \left (\frac {-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {1}{3}} x^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}+1\right )^{\frac {2}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{x^{6}-x^{3}+1}\right )-\ln \left (-\frac {-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}+1\right )^{\frac {2}{3}} x -9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-2 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{x^{6}-x^{3}+1}\right )\) | \(435\) |
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} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{3} {\left (x^{3} - 1\right )}}{{\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 )\,{\left (x^3+1\right )}^3\,{\left (x^6+1\right )}^{2/3}}{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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