Optimal. Leaf size=85 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )-\log \left (x+\sqrt [3]{x^2+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.70, antiderivative size = 297, normalized size of antiderivative = 3.49, number of steps
used = 18, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2081, 6860,
371, 973, 477, 440, 476, 524} \begin {gather*} \frac {3 \sqrt [3]{x^2+1} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};-\frac {4 x^2}{\left (i-\sqrt {3}\right )^2},-x^2\right )}{2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x^4+x^2}}+\frac {3 \sqrt [3]{x^2+1} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};-\frac {4 x^2}{\left (i+\sqrt {3}\right )^2},-x^2\right )}{2 \left (1-i \sqrt {3}\right ) \sqrt [3]{x^4+x^2}}-\frac {3 \sqrt [3]{x^2+1} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {4 x^2}{\left (i-\sqrt {3}\right )^2},-x^2\right )}{\sqrt [3]{x^4+x^2}}-\frac {3 \sqrt [3]{x^2+1} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {4 x^2}{\left (i+\sqrt {3}\right )^2},-x^2\right )}{\sqrt [3]{x^4+x^2}}+\frac {3 \sqrt [3]{x^2+1} x \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 440
Rule 476
Rule 477
Rule 524
Rule 973
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {-1+x^2}{x^{2/3} \sqrt [3]{1+x^2} \left (1+x+x^2\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (\frac {1}{x^{2/3} \sqrt [3]{1+x^2}}-\frac {2+x}{x^{2/3} \sqrt [3]{1+x^2} \left (1+x+x^2\right )}\right ) \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {2+x}{x^{2/3} \sqrt [3]{1+x^2} \left (1+x+x^2\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (\frac {1-i \sqrt {3}}{x^{2/3} \left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^2}}+\frac {1+i \sqrt {3}}{x^{2/3} \left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^2}}\right ) \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (\left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1}{x^{2/3} \left (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}-\frac {\left (\left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1}{x^{2/3} \left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {\sqrt [3]{x}}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}-\frac {\left (\left (1-i \sqrt {3}\right )^2 x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1}{x^{2/3} \left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {\sqrt [3]{x}}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}-\frac {\left (\left (1+i \sqrt {3}\right )^2 x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1}{x^{2/3} \left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\left (\left (1-i \sqrt {3}\right )^2-4 x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (1-i \sqrt {3}\right )^2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\left (\left (1+i \sqrt {3}\right )^2-4 x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (1+i \sqrt {3}\right )^2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {3 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {4 x^2}{\left (i-\sqrt {3}\right )^2},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {3 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {4 x^2}{\left (i+\sqrt {3}\right )^2},-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\left (\left (1-i \sqrt {3}\right )^2-4 x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {3 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {4 x^2}{\left (i-\sqrt {3}\right )^2},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {3 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {4 x^2}{\left (i+\sqrt {3}\right )^2},-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x^2 \sqrt [3]{1+x^2} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};-\frac {4 x^2}{\left (i-\sqrt {3}\right )^2},-x^2\right )}{2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x^2+x^4}}+\frac {3 x^2 \sqrt [3]{1+x^2} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};-\frac {4 x^2}{\left (i+\sqrt {3}\right )^2},-x^2\right )}{2 \left (1-i \sqrt {3}\right ) \sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 118, normalized size = 1.39 \begin {gather*} \frac {x^{2/3} \sqrt [3]{1+x^2} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+\log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{2 \sqrt [3]{x^2+x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.85, size = 371, normalized size = 4.36
method | result | size |
trager | \(\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 x^{3}-3 \left (x^{4}+x^{2}\right )^{\frac {2}{3}}+3 x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -x^{2}+2 x}{\left (x^{2}+x +1\right ) x}\right )-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -x^{2}-x}{\left (x^{2}+x +1\right ) x}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -x^{2}-x}{\left (x^{2}+x +1\right ) x}\right )\) | \(371\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 106, normalized size = 1.25 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + 4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{8 \, x^{3} - x^{2} + 8 \, x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{3} + x^{2} + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} + x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + x + 1\right )}\, dx \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*} \text {could not integrate} \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 {x^2-1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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