Optimal. Leaf size=82 \[ \log \left (\sqrt [3]{x^6+x^2}-x\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^6+x^2} x+\left (x^6+x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+x^2}+x}\right ) \]
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Rubi [F] time = 1.06, 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 {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 (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 = 2.66, size = 82, 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 = 1.81, size = 112, normalized size = 1.37 \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 = 4.34, size = 511, normalized size = 6.23
method | result | size |
trager | \(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}-1101 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+9494 x^{5}+3646 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x -1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +14963 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+21912 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+21912 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}-1101 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +14241 x^{2}+9494 x}{x \left (x^{4}-x +1\right )}\right )-\ln \left (\frac {1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-8772 x^{5}-3646 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x +1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +7671 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9873 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-9873 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2924 x^{2}-8772 x}{x \left (x^{4}-x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-8772 x^{5}-3646 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {2}{3}}+12039 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{3}} x +1823 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +7671 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9873 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}-9873 x \left (x^{6}+x^{2}\right )^{\frac {1}{3}}+2545 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2924 x^{2}-8772 x}{x \left (x^{4}-x +1\right )}\right )\) | \(511\) |
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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