Optimal. Leaf size=86 \[ \log \left (\sqrt [3]{x^2+1}-x\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^2+1} x+\left (x^2+1\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {2 x}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right ) \]
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Rubi [F] time = 0.44, 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 {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx &=\int \left (\frac {3}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )}+\frac {x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )}\right ) \, dx\\ &=3 \int \frac {1}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx+\int \frac {x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.13, size = 86, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )+\log \left (-x+\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 104, normalized size = 1.21 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{3} - 2 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 8 \, x^{2} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} - x^{2} + 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} x - 1}{x^{3} - x^{2} - 1}\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 {x^{2} + 3}{{\left (x^{3} - x^{2} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.19, size = 287, normalized size = 3.34
method | result | size |
trager | \(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x +\left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 \left (x^{2}+1\right )^{\frac {2}{3}} x +2 \left (x^{2}+1\right )^{\frac {1}{3}} x^{2}+x^{3}+x^{2}+1}{x^{3}-x^{2}-1}\right )-\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x +\left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\left (x^{2}+1\right )^{\frac {2}{3}} x -\left (x^{2}+1\right )^{\frac {1}{3}} x^{2}-x^{2}-1}{x^{3}-x^{2}-1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x +\left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\left (x^{2}+1\right )^{\frac {2}{3}} x -\left (x^{2}+1\right )^{\frac {1}{3}} x^{2}-x^{2}-1}{x^{3}-x^{2}-1}\right )\) | \(287\) |
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 {x^{2} + 3}{{\left (x^{3} - x^{2} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{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 {x^2+3}{{\left (x^2+1\right )}^{1/3}\,\left (-x^3+x^2+1\right )} \,d x \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 {x^{2} + 3}{\sqrt [3]{x^{2} + 1} \left (x^{3} - x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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