Optimal. Leaf size=38 \[ \frac {\sqrt {-1+x^3}}{x}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^3}}\right ) \]
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Rubi [F]
time = 0.46, 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 {\sqrt {-1+x^3} \left (2+x^3\right )}{x^2 \left (-2-4 x^2+2 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^3} \left (2+x^3\right )}{x^2 \left (-2-4 x^2+2 x^3\right )} \, dx &=\int \left (-\frac {\sqrt {-1+x^3}}{x^2}+\frac {(-4+3 x) \sqrt {-1+x^3}}{2 \left (-1-2 x^2+x^3\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {(-4+3 x) \sqrt {-1+x^3}}{-1-2 x^2+x^3} \, dx-\int \frac {\sqrt {-1+x^3}}{x^2} \, dx\\ &=\frac {\sqrt {-1+x^3}}{x}+\frac {1}{2} \int \left (-\frac {4 \sqrt {-1+x^3}}{-1-2 x^2+x^3}+\frac {3 x \sqrt {-1+x^3}}{-1-2 x^2+x^3}\right ) \, dx-\frac {3}{2} \int \frac {x}{\sqrt {-1+x^3}} \, dx\\ &=\frac {\sqrt {-1+x^3}}{x}+\frac {3}{2} \int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx+\frac {3}{2} \int \frac {x \sqrt {-1+x^3}}{-1-2 x^2+x^3} \, dx-2 \int \frac {\sqrt {-1+x^3}}{-1-2 x^2+x^3} \, dx-\left (3 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx\\ &=\frac {3 \sqrt {-1+x^3}}{1-\sqrt {3}-x}+\frac {\sqrt {-1+x^3}}{x}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {2} 3^{3/4} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {3}{2} \int \frac {x \sqrt {-1+x^3}}{-1-2 x^2+x^3} \, dx-2 \int \frac {\sqrt {-1+x^3}}{-1-2 x^2+x^3} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 38, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^3}}{x}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^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 = 1.64, size = 306, normalized size = 8.05
method | result | size |
trager | \(\frac {\sqrt {x^{3}-1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{3}-1}\, x -\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{3}-2 x^{2}-1}\right )}{2}\) | \(76\) |
default | \(\frac {\sqrt {x^{3}-1}}{x}+\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-2 \textit {\_Z}^{2}-1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{4}+\frac {3 \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {3}{4}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{4}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{2}\) | \(306\) |
risch | \(\frac {\sqrt {x^{3}-1}}{x}+\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-2 \textit {\_Z}^{2}-1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{4}+\frac {3 \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {3}{4}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{4}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{2}\) | \(306\) |
elliptic | \(\frac {\sqrt {x^{3}-1}}{x}+\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-2 \textit {\_Z}^{2}-1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {-1+x}{-3-i \sqrt {3}}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{4}+\frac {3 \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {3}{4}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{4}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{2}\) | \(306\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (30) = 60\).
time = 0.41, size = 99, normalized size = 2.61 \begin {gather*} \frac {\sqrt {2} x \log \left (-\frac {x^{6} + 12 \, x^{5} + 4 \, x^{4} - 2 \, x^{3} - 4 \, \sqrt {2} {\left (x^{4} + 2 \, x^{3} - x\right )} \sqrt {x^{3} - 1} - 12 \, x^{2} + 1}{x^{6} - 4 \, x^{5} + 4 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1}\right ) + 4 \, \sqrt {x^{3} - 1}}{4 \, 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*} \frac {\int \frac {2 \sqrt {x^{3} - 1}}{x^{5} - 2 x^{4} - x^{2}}\, dx + \int \frac {x^{3} \sqrt {x^{3} - 1}}{x^{5} - 2 x^{4} - x^{2}}\, dx}{2} \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 [B]
time = 0.82, size = 56, normalized size = 1.47 \begin {gather*} \frac {\sqrt {x^3-1}}{x}+\frac {\sqrt {2}\,\ln \left (\frac {2\,x^2+x^3-2\,\sqrt {2}\,x\,\sqrt {x^3-1}-1}{-8\,x^3+16\,x^2+8}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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