Optimal. Leaf size=94 \[ \frac {1}{3} \log \left (\sqrt [3]{x^6+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+1}+x}\right )}{\sqrt {3}}+\frac {\left (x^6+1\right )^{2/3}}{2 x^2}-\frac {1}{6} \log \left (\sqrt [3]{x^6+1} x+\left (x^6+1\right )^{2/3}+x^2\right ) \]
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Rubi [C] time = 0.79, antiderivative size = 247, normalized size of antiderivative = 2.63, number of steps used = 16, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6728, 275, 364, 1438, 429, 465, 510} \begin {gather*} \frac {\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 {\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 {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 {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 {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 275
Rule 364
Rule 429
Rule 465
Rule 510
Rule 1438
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx &=\int \left (-\frac {\left (1+x^6\right )^{2/3}}{x^3}+\frac {\left (-1+2 x^3\right ) \left (1+x^6\right )^{2/3}}{1-x^3+x^6}\right ) \, dx\\ &=-\int \frac {\left (1+x^6\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1+2 x^3\right ) \left (1+x^6\right )^{2/3}}{1-x^3+x^6} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+\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 {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+2 \int \frac {\left (1+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^3} \, dx+2 \int \frac {\left (1+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^3} \, dx\\ &=\frac {\, _2F_1\left (-\frac {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+2 \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+2 \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 {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+2 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{1-i \sqrt {3}+2 x^6} \, dx+2 \int \frac {x^3 \left (1+x^6\right )^{2/3}}{1+i \sqrt {3}+2 x^6} \, dx+\left (-i-\sqrt {3}\right ) \int \frac {\left (1+x^6\right )^{2/3}}{-i+\sqrt {3}-2 i x^6} \, dx+\left (i-\sqrt {3}\right ) \int \frac {\left (1+x^6\right )^{2/3}}{i+\sqrt {3}+2 i x^6} \, dx\\ &=\frac {\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 {\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 {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}+\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 )+\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 {\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 {\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 {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 {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 {2}{3},-\frac {1}{3};\frac {2}{3};-x^6\right )}{2 x^2}\\ \end {align*}
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Mathematica [F] time = 1.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (1-x^3+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.91, size = 94, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (\sqrt [3]{x^6+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+1}+x}\right )}{\sqrt {3}}+\frac {\left (x^6+1\right )^{2/3}}{2 x^2}-\frac {1}{6} \log \left (\sqrt [3]{x^6+1} x+\left (x^6+1\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 9.02, size = 135, normalized size = 1.44 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \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 ) - x^{2} \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 ) - 3 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \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^{6} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.90, size = 282, normalized size = 3.00 \begin {gather*} \frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (\frac {3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-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}+3 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}-x^{3}+1}\right )}{3}+\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}+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 -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}+1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{6}+1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}-x^{3}+1}\right ) \end {gather*}
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^{6} - 1\right )}}{{\left (x^{6} - x^{3} + 1\right )} x^{3}}\,{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^6-1\right )\,{\left (x^6+1\right )}^{2/3}}{x^3\,\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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