Optimal. Leaf size=92 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \left (-1+x^2\right )^{2/3}}{-2-2 x+\left (-1+x^2\right )^{2/3}}\right )-\log \left (1+x+\left (-1+x^2\right )^{2/3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(-1-x) \left (-1+x^2\right )^{2/3}+\left (-1+x^2\right )^{4/3}\right ) \]
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Rubi [F]
time = 0.25, 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+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(-1+x) (3+x)}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx &=\int \left (\frac {1}{\left (-1+x^2\right )^{2/3}}-\frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )}\right ) \, dx\\ &=\int \frac {1}{\left (-1+x^2\right )^{2/3}} \, dx-\int \frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\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 \frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\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}}}-\int \frac {5-3 x}{\left (-1+x^2\right )^{2/3} \left (2-x+x^2\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 92, normalized size = 1.00 \begin {gather*} -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \left (-1+x^2\right )^{2/3}}{-2-2 x+\left (-1+x^2\right )^{2/3}}\right )-\log \left (1+x+\left (-1+x^2\right )^{2/3}\right )+\frac {1}{2} \log \left (1+2 x+x^2+(-1-x) \left (-1+x^2\right )^{2/3}+\left (-1+x^2\right )^{4/3}\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 = 1.80, size = 315, normalized size = 3.42
method | result | size |
trager | \(-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}+2 x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-3 \left (x^{2}-1\right )^{\frac {1}{3}}-4 x +2}{x^{2}-x +2}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+7 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +3 \left (x^{2}-1\right )^{\frac {2}{3}}+3 x \left (x^{2}-1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-3 \left (x^{2}-1\right )^{\frac {1}{3}}-2 x -2}{x^{2}-x +2}\right )\) | \(315\) |
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 = 0.73, size = 98, normalized size = 1.07 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, x^{2} - 17 \, x + 7}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} + 3 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - x + 3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 2}{x^{2} - 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 {\left (x - 1\right ) \left (x + 3\right )}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {2}{3}} \left (x^{2} - x + 2\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 {\left (x-1\right )\,\left (x+3\right )}{{\left (x^2-1\right )}^{2/3}\,\left (x^2-x+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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