Optimal. Leaf size=21 \[ x-\frac {e^{x+x^2}}{\frac {x}{2}+\log (2)} \]
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Rubi [A] time = 0.32, antiderivative size = 38, normalized size of antiderivative = 1.81, number of steps used = 4, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {27, 6688, 2288} \begin {gather*} x-\frac {2 e^{x^2+x} \left (2 x^2+x (1+\log (16))+\log (4)\right )}{(2 x+1) (x+\log (4))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 2288
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2+4 x \log (2)+4 \log ^2(2)+e^{x+x^2} \left (2-2 x-4 x^2+(-4-8 x) \log (2)\right )}{(x+2 \log (2))^2} \, dx\\ &=\int \left (1-\frac {2 e^{x+x^2} \left (-1+2 x^2+\log (4)+x (1+\log (16))\right )}{(x+\log (4))^2}\right ) \, dx\\ &=x-2 \int \frac {e^{x+x^2} \left (-1+2 x^2+\log (4)+x (1+\log (16))\right )}{(x+\log (4))^2} \, dx\\ &=x-\frac {2 e^{x+x^2} \left (2 x^2+\log (4)+x (1+\log (16))\right )}{(1+2 x) (x+\log (4))^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 38, normalized size = 1.81 \begin {gather*} x-\frac {2 e^{x+x^2} \left (2 x^2+\log (4)+x (1+\log (16))\right )}{(1+2 x) (x+\log (4))^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 26, normalized size = 1.24 \begin {gather*} \frac {x^{2} + 2 \, x \log \relax (2) - 2 \, e^{\left (x^{2} + x\right )}}{x + 2 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 26, normalized size = 1.24 \begin {gather*} \frac {x^{2} + 2 \, x \log \relax (2) - 2 \, e^{\left (x^{2} + x\right )}}{x + 2 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 19, normalized size = 0.90
method | result | size |
risch | \(x -\frac {2 \,{\mathrm e}^{\left (x +1\right ) x}}{x +2 \ln \relax (2)}\) | \(19\) |
norman | \(\frac {x^{2}-2 \,{\mathrm e}^{x^{2}+x}-4 \ln \relax (2)^{2}}{x +2 \ln \relax (2)}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 67, normalized size = 3.19 \begin {gather*} 4 \, {\left (\frac {2 \, \log \relax (2)}{x + 2 \, \log \relax (2)} + \log \left (x + 2 \, \log \relax (2)\right )\right )} \log \relax (2) - 4 \, \log \relax (2) \log \left (x + 2 \, \log \relax (2)\right ) + x - \frac {8 \, \log \relax (2)^{2}}{x + 2 \, \log \relax (2)} - \frac {2 \, e^{\left (x^{2} + x\right )}}{x + 2 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 16, normalized size = 0.76 \begin {gather*} x-\frac {2\,{\mathrm {e}}^{x^2+x}}{x+\ln \relax (4)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 15, normalized size = 0.71 \begin {gather*} x - \frac {2 e^{x^{2} + x}}{x + 2 \log {\relax (2 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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