Optimal. Leaf size=32 \[ -e^{x-x^2}-\frac {4}{-4+x}-5 x+\frac {x}{2 (2+x)} \]
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Rubi [A] time = 0.25, antiderivative size = 33, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {6688, 2236, 1660, 967, 8} \begin {gather*} -e^{x-x^2}+\frac {5 x+4}{-x^2+2 x+8}-5 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 967
Rule 1660
Rule 2236
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{x-x^2} (-1+2 x)+\frac {-288-152 x+65 x^2+20 x^3-5 x^4}{\left (-8-2 x+x^2\right )^2}\right ) \, dx\\ &=\int e^{x-x^2} (-1+2 x) \, dx+\int \frac {-288-152 x+65 x^2+20 x^3-5 x^4}{\left (-8-2 x+x^2\right )^2} \, dx\\ &=-e^{x-x^2}+\frac {4+5 x}{8+2 x-x^2}-\frac {1}{36} \int \frac {-1440-360 x+180 x^2}{-8-2 x+x^2} \, dx\\ &=-e^{x-x^2}+\frac {4+5 x}{8+2 x-x^2}-5 \int 1 \, dx\\ &=-e^{x-x^2}-5 x+\frac {4+5 x}{8+2 x-x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 31, normalized size = 0.97 \begin {gather*} -e^{x-x^2}-5 x+\frac {-4-5 x}{-8-2 x+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 44, normalized size = 1.38 \begin {gather*} -\frac {5 \, x^{3} - 10 \, x^{2} + {\left (x^{2} - 2 \, x - 8\right )} e^{\left (-x^{2} + x\right )} - 35 \, x + 4}{x^{2} - 2 \, x - 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 60, normalized size = 1.88 \begin {gather*} -\frac {5 \, x^{3} + x^{2} e^{\left (-x^{2} + x\right )} - 10 \, x^{2} - 2 \, x e^{\left (-x^{2} + x\right )} - 35 \, x - 8 \, e^{\left (-x^{2} + x\right )} + 4}{x^{2} - 2 \, x - 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 29, normalized size = 0.91
method | result | size |
default | \(-{\mathrm e}^{-x^{2}+x}-\frac {1}{2+x}-\frac {4}{x -4}-5 x\) | \(29\) |
risch | \(-5 x +\frac {-5 x -4}{x^{2}-2 x -8}-{\mathrm e}^{-x \left (x -1\right )}\) | \(30\) |
norman | \(\frac {55 x -5 x^{3}+2 \,{\mathrm e}^{-x^{2}+x} x -{\mathrm e}^{-x^{2}+x} x^{2}+8 \,{\mathrm e}^{-x^{2}+x}+76}{x^{2}-2 x -8}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 95, normalized size = 2.97 \begin {gather*} -5 \, x + \frac {20 \, {\left (17 \, x + 28\right )}}{9 \, {\left (x^{2} - 2 \, x - 8\right )}} - \frac {40 \, {\left (7 \, x + 20\right )}}{9 \, {\left (x^{2} - 2 \, x - 8\right )}} - \frac {65 \, {\left (5 \, x + 4\right )}}{9 \, {\left (x^{2} - 2 \, x - 8\right )}} + \frac {76 \, {\left (x + 8\right )}}{9 \, {\left (x^{2} - 2 \, x - 8\right )}} + \frac {16 \, {\left (x - 1\right )}}{x^{2} - 2 \, x - 8} - e^{\left (-x^{2} + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 32, normalized size = 1.00 \begin {gather*} \frac {5\,x+4}{-x^2+2\,x+8}-{\mathrm {e}}^{x-x^2}-5\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 24, normalized size = 0.75 \begin {gather*} - 5 x - \frac {5 x + 4}{x^{2} - 2 x - 8} - e^{- x^{2} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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