Optimal. Leaf size=22 \[ -\frac {3 \left (x^3-x\right )^{2/3}}{2 \left (x^2-1\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 16, normalized size of antiderivative = 0.73, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2056, 264} \begin {gather*} -\frac {3 x}{2 \sqrt [3]{x^3-x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^2\right ) \sqrt [3]{-x+x^3}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \left (-1+x^2\right )^{4/3}} \, dx}{\sqrt [3]{-x+x^3}}\\ &=-\frac {3 x}{2 \sqrt [3]{-x+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.73 \begin {gather*} -\frac {3 x}{2 \sqrt [3]{x \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 22, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-x+x^3\right )^{2/3}}{2 \left (-1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 11, normalized size = 0.50 \begin {gather*} -\frac {3}{2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 13, normalized size = 0.59
method | result | size |
gosper | \(-\frac {3 x}{2 \left (x^{3}-x \right )^{\frac {1}{3}}}\) | \(13\) |
risch | \(-\frac {3 x}{2 \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}\) | \(13\) |
trager | \(-\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}}}{2 \left (x^{2}-1\right )}\) | \(19\) |
meijerg | \(-\frac {3 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {2}{3}}}{2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{3}}}\) | \(33\) |
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 {1}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 20, normalized size = 0.91 \begin {gather*} -\frac {3\,{\left (x^3-x\right )}^{2/3}}{2\,\left (x^2-1\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 {1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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