Optimal. Leaf size=20 \[ -\frac {2 \sqrt {x^3-x}}{x^2-1} \]
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Rubi [A] time = 0.08, antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2056, 449} \begin {gather*} -\frac {2 x}{\sqrt {x^3-x}} \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 {-x+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1+x^2}{\sqrt {x} \left (-1+x^2\right )^{3/2}} \, dx}{\sqrt {-x+x^3}}\\ &=-\frac {2 x}{\sqrt {-x+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 14, normalized size = 0.70 \begin {gather*} -\frac {2 x}{\sqrt {x \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 20, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {-x+x^3}}{-1+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {x^{3} - x}}{x^{2} - 1} \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}{\sqrt {x^{3} - x} {\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 = 0.12, size = 13, normalized size = 0.65
method | result | size |
gosper | \(-\frac {2 x}{\sqrt {x^{3}-x}}\) | \(13\) |
risch | \(-\frac {2 x}{\sqrt {x \left (x^{2}-1\right )}}\) | \(13\) |
elliptic | \(-\frac {2 x}{\sqrt {x \left (x^{2}-1\right )}}\) | \(13\) |
trager | \(-\frac {2 \sqrt {x^{3}-x}}{x^{2}-1}\) | \(19\) |
default | \(\frac {x^{2}-x}{\sqrt {\left (1+x \right ) \left (x^{2}-x \right )}}-\frac {x^{2}+x}{\sqrt {\left (-1+x \right ) \left (x^{2}+x \right )}}\) | \(41\) |
meijerg | \(-\frac {2 \sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \hypergeom \left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], x^{2}\right ) \sqrt {x}}{\sqrt {\mathrm {signum}\left (x^{2}-1\right )}}-\frac {2 \sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \hypergeom \left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], x^{2}\right ) x^{\frac {5}{2}}}{5 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}\) | \(66\) |
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}{\sqrt {x^{3} - x} {\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 = 0.07, size = 12, normalized size = 0.60 \begin {gather*} -\frac {2\,x}{\sqrt {x^3-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*} \int \frac {x^{2} + 1}{\sqrt {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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