Optimal. Leaf size=27 \[ \frac {e^{-6+\frac {1}{2} e^2 (-5+x)+2 x^2}}{-1+x}+x \]
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Rubi [B] time = 0.44, antiderivative size = 65, normalized size of antiderivative = 2.41, number of steps used = 5, number of rules used = 4, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {27, 12, 6742, 2288} \begin {gather*} x-\frac {e^{2 x^2+\frac {e^2 x}{2}+\frac {1}{2} \left (-12-5 e^2\right )} \left (-8 x^2+\left (8-e^2\right ) x+e^2\right )}{(1-x)^2 \left (8 x+e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 2288
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2-4 x+2 x^2+e^{\frac {1}{2} \left (-12+e^2 (-5+x)+4 x^2\right )} \left (-2+e^2 (-1+x)-8 x+8 x^2\right )}{2 (-1+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {2-4 x+2 x^2+e^{\frac {1}{2} \left (-12+e^2 (-5+x)+4 x^2\right )} \left (-2+e^2 (-1+x)-8 x+8 x^2\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{2} \int \left (2+\frac {e^{\frac {1}{2} \left (-12-5 e^2\right )+\frac {e^2 x}{2}+2 x^2} \left (-2-e^2-\left (8-e^2\right ) x+8 x^2\right )}{(1-x)^2}\right ) \, dx\\ &=x+\frac {1}{2} \int \frac {e^{\frac {1}{2} \left (-12-5 e^2\right )+\frac {e^2 x}{2}+2 x^2} \left (-2-e^2-\left (8-e^2\right ) x+8 x^2\right )}{(1-x)^2} \, dx\\ &=x-\frac {e^{\frac {1}{2} \left (-12-5 e^2\right )+\frac {e^2 x}{2}+2 x^2} \left (e^2+\left (8-e^2\right ) x-8 x^2\right )}{(1-x)^2 \left (e^2+8 x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 27, normalized size = 1.00 \begin {gather*} \frac {e^{-6+\frac {1}{2} e^2 (-5+x)+2 x^2}}{-1+x}+x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 28, normalized size = 1.04 \begin {gather*} \frac {x^{2} - x + e^{\left (2 \, x^{2} + \frac {1}{2} \, {\left (x - 5\right )} e^{2} - 6\right )}}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 41, normalized size = 1.52 \begin {gather*} \frac {x^{2} e^{2} - x e^{2} + e^{\left (2 \, x^{2} + \frac {1}{2} \, x e^{2} - \frac {5}{2} \, e^{2} - 4\right )}}{x e^{2} - e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 26, normalized size = 0.96
method | result | size |
risch | \(\frac {{\mathrm e}^{2 x^{2}+\frac {{\mathrm e}^{2} x}{2}-\frac {5 \,{\mathrm e}^{2}}{2}-6}}{x -1}+x\) | \(26\) |
norman | \(\frac {x^{2}+{\mathrm e}^{\frac {\left (x -5\right ) {\mathrm e}^{2}}{2}+2 x^{2}-6}-1}{x -1}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 36, normalized size = 1.33 \begin {gather*} x + \frac {e^{\left (2 \, x^{2} + \frac {1}{2} \, x e^{2}\right )}}{x e^{\left (\frac {5}{2} \, e^{2} + 6\right )} - e^{\left (\frac {5}{2} \, e^{2} + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 25, normalized size = 0.93 \begin {gather*} x+\frac {{\mathrm {e}}^{2\,x^2+\frac {{\mathrm {e}}^2\,x}{2}-\frac {5\,{\mathrm {e}}^2}{2}-6}}{x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 24, normalized size = 0.89 \begin {gather*} x + \frac {e^{2 x^{2} + 2 \left (\frac {x}{4} - \frac {5}{4}\right ) e^{2} - 6}}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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