Optimal. Leaf size=93 \[ -\log \left (\sqrt [3]{x^6-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^2} x+\left (x^6-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x^2}-x}\right ) \]
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Rubi [F] time = 0.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^4}\right ) \int \frac {1+3 x^4}{x^{2/3} \sqrt [3]{-1+x^4} \left (-1+x+x^4\right )} \, dx}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+3 x^{12}}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt [3]{-1+x^{12}}}+\frac {4-3 x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {4-3 x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}+\frac {\left (9 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^{12}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {\left (9 x^{2/3} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^{12}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}+\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {4}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )}-\frac {3 x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ &=\frac {9 x \sqrt [3]{1-x^4} \, _2F_1\left (\frac {1}{12},\frac {1}{3};\frac {13}{12};x^4\right )}{\sqrt [3]{-x^2+x^6}}-\frac {\left (9 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}+\frac {\left (12 x^{2/3} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^{12}} \left (-1+x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^6}}\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.62, size = 93, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x^2+x^6}}\right )-\log \left (x+\sqrt [3]{-x^2+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x^2+x^6}+\left (-x^2+x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.89, size = 124, normalized size = 1.33 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} + x^{2} - x\right )} + 2 \, \sqrt {3} {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{5} - x^{2} - x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} + x^{2} + 3 \, {\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{x^{5} + x^{2} - x}\right ) \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 {3 \, x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{4} + x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.37, size = 386, normalized size = 4.15
method | result | size |
trager | \(-\ln \left (-\frac {-680618 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}+11776766 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}-23512440 x^{5}+5104635 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+10436076 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-10374438 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{3}} x +680618 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -20604254 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-41621028 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-20872152 x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}-11776766 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +20377448 x^{2}+23512440 x}{x \left (x^{4}+x -1\right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {-391874 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}-4413744 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}+8848034 x^{5}+2939055 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+5218038 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}+10405257 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{3}} x +391874 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -5888383 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+10374438 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-10436076 x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}+4413744 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +1361236 x^{2}-8848034 x}{x \left (x^{4}+x -1\right )}\right )}{2}\) | \(386\) |
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 {3 \, x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{4} + x - 1\right )}}\,{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 {3\,x^4+1}{{\left (x^6-x^2\right )}^{1/3}\,\left (x^4+x-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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