Optimal. Leaf size=21 \[ \frac {2 \sqrt {2+x-x^2}}{3 (-2+x)} \]
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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps
used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {664}
\begin {gather*} -\frac {2 \sqrt {-x^2+x+2}}{3 (2-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rubi steps
\begin {align*} \int \frac {1}{(-2+x) \sqrt {2+x-x^2}} \, dx &=-\frac {2 \sqrt {2+x-x^2}}{3 (2-x)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 21, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {2+x-x^2}}{3 (-2+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 22, normalized size = 1.05
method | result | size |
gosper | \(-\frac {2 \left (1+x \right )}{3 \sqrt {-x^{2}+x +2}}\) | \(16\) |
risch | \(-\frac {2 \left (1+x \right )}{3 \sqrt {-x^{2}+x +2}}\) | \(16\) |
trager | \(\frac {2 \sqrt {-x^{2}+x +2}}{3 \left (-2+x \right )}\) | \(18\) |
default | \(\frac {2 \sqrt {-\left (-2+x \right )^{2}+6-3 x}}{3 \left (-2+x \right )}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.86, size = 17, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + x + 2}}{3 \, {\left (x - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 17, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + x + 2}}{3 \, {\left (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 {1}{\sqrt {- \left (x - 2\right ) \left (x + 1\right )} \left (x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 28, normalized size = 1.33 \begin {gather*} -\frac {4}{3 \, {\left (\frac {2 \, \sqrt {-x^{2} + x + 2} - 3}{2 \, x - 1} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 19, normalized size = 0.90 \begin {gather*} \frac {2\,\sqrt {-x^2+x+2}}{3\,\left (x-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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