Optimal. Leaf size=25 \[ -\frac {3 \left (x^4-x^2\right )^{2/3}}{x \left (x^2-1\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 16, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2056, 449} \begin {gather*} -\frac {3 x}{\sqrt [3]{x^4-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{-x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {1+x^2}{x^{2/3} \left (-1+x^2\right )^{4/3}} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=-\frac {3 x}{\sqrt [3]{-x^2+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 0.64 \begin {gather*} -\frac {3 x}{\sqrt [3]{x^2 \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 25, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-x^2+x^4\right )^{2/3}}{x \left (-1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 22, normalized size = 0.88 \begin {gather*} -\frac {3 \, {\left (x^{4} - x^{2}\right )}^{\frac {2}{3}}}{x^{3} - x} \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} + 1}{{\left (x^{4} - x^{2}\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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maple [A] time = 0.10, size = 15, normalized size = 0.60
method | result | size |
gosper | \(-\frac {3 x}{\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\) | \(15\) |
risch | \(-\frac {3 x}{\left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{3}}}\) | \(15\) |
trager | \(-\frac {3 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x \left (x^{2}-1\right )}\) | \(24\) |
meijerg | \(-\frac {3 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{6}, \frac {4}{3}\right ], \left [\frac {7}{6}\right ], x^{2}\right ) x^{\frac {1}{3}}}{\mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}-\frac {3 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} \hypergeom \left (\left [\frac {7}{6}, \frac {4}{3}\right ], \left [\frac {13}{6}\right ], x^{2}\right ) x^{\frac {7}{3}}}{7 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}}}\) | \(66\) |
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} + 1}{{\left (x^{4} - x^{2}\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 = 0.19, size = 23, normalized size = 0.92 \begin {gather*} -\frac {3\,{\left (x^4-x^2\right )}^{2/3}}{x\,\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 {x^{2} + 1}{\sqrt [3]{x^{2} \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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