Optimal. Leaf size=73 \[ \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^2}}\right )+\log \left (-x+\sqrt [3]{-1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right ) \]
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Rubi [F]
time = 0.56, 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 {x \left (-3+x^2\right )}{\left (-1+x^2\right )^{2/3} \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 {x \left (-3+x^2\right )}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx &=\int \left (\frac {1}{\left (-1+x^2\right )^{2/3}}-\frac {1+3 x-x^2}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )}\right ) \, dx\\ &=\int \frac {1}{\left (-1+x^2\right )^{2/3}} \, dx-\int \frac {1+3 x-x^2}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx\\ &=\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}-\int \left (\frac {1}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )}+\frac {3 x}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )}-\frac {x^2}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )}\right ) \, dx\\ &=\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right )|-7-4 \sqrt {3}\right )}{x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}-3 \int \frac {x}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx-\int \frac {1}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx+\int \frac {x^2}{\left (-1+x^2\right )^{2/3} \left (1-x^2+x^3\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.26, size = 73, normalized size = 1.00 \begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.78, size = 224, normalized size = 3.07
method | result | size |
trager | \(\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-x \left (x^{2}-1\right )^{\frac {2}{3}}+2 \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{x^{3}-x^{2}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}-1\right )^{\frac {2}{3}}-3 \left (x^{2}-1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+x^{3}+x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{3}-x^{2}+1}\right )\) | \(224\) |
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 = 0.36, size = 74, normalized size = 1.01 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \log \left (-\frac {x - {\left (x^{2} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{2} - 1\right )}^{\frac {1}{3}} x + {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{x^{2}}\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 {x \left (x^{2} - 3\right )}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - x^{2} + 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\,\left (x^2-3\right )}{{\left (x^2-1\right )}^{2/3}\,\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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