Optimal. Leaf size=29 \[ 5-\frac {4}{x}-\left (2+\frac {x}{2}\right ) \left (1+e-4 e^{-e^2}+x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6, 12, 14} \begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \left (5+e-4 e^{-e^2}\right ) x-\frac {4}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8+(-5-e) x^2+4 e^{-e^2} x^2-2 x^3}{2 x^2} \, dx\\ &=\int \frac {8+\left (-5-e+4 e^{-e^2}\right ) x^2-2 x^3}{2 x^2} \, dx\\ &=\frac {1}{2} \int \frac {8+\left (-5-e+4 e^{-e^2}\right ) x^2-2 x^3}{x^2} \, dx\\ &=\frac {1}{2} \int \left (-5 \left (1+\frac {1}{5} \left (e-4 e^{-e^2}\right )\right )+\frac {8}{x^2}-2 x\right ) \, dx\\ &=-\frac {4}{x}-\frac {1}{2} \left (5+e-4 e^{-e^2}\right ) x-\frac {x^2}{2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 1.17 \begin {gather*} -\frac {4}{x}-\frac {5 x}{2}-\frac {e x}{2}+2 e^{-e^2} x-\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 36, normalized size = 1.24 \begin {gather*} -\frac {x^{3} + x^{2} e - x^{2} e^{\left (-e^{2} + 2 \, \log \relax (2)\right )} + 5 \, x^{2} + 8}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 32, normalized size = 1.10 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, x e + \frac {1}{2} \, x e^{\left (-e^{2} + 2 \, \log \relax (2)\right )} - \frac {5}{2} \, x - \frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 33, normalized size = 1.14
method | result | size |
default | \(-\frac {x^{2}}{2}-\frac {x \,{\mathrm e}}{2}+\frac {x \,{\mathrm e}^{2 \ln \relax (2)-{\mathrm e}^{2}}}{2}-\frac {5 x}{2}-\frac {4}{x}\) | \(33\) |
norman | \(\frac {-4-\frac {x^{3}}{2}-\frac {{\mathrm e}^{-{\mathrm e}^{2}} \left ({\mathrm e} \,{\mathrm e}^{{\mathrm e}^{2}}+5 \,{\mathrm e}^{{\mathrm e}^{2}}-4\right ) x^{2}}{2}}{x}\) | \(35\) |
gosper | \(-\frac {x^{2} {\mathrm e}-x^{2} {\mathrm e}^{2 \ln \relax (2)-{\mathrm e}^{2}}+x^{3}+5 x^{2}+8}{2 x}\) | \(37\) |
risch | \(-\frac {{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{-{\mathrm e}^{2}} x^{2}}{2}-\frac {{\mathrm e}^{{\mathrm e}^{2}+1} {\mathrm e}^{-{\mathrm e}^{2}} x}{2}-\frac {5 \,{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{-{\mathrm e}^{2}} x}{2}+2 \,{\mathrm e}^{-{\mathrm e}^{2}} x -\frac {4}{x}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 33, normalized size = 1.14 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} e^{\left (e^{2}\right )} + {\left ({\left (e + 5\right )} e^{\left (e^{2}\right )} - 4\right )} x\right )} e^{\left (-e^{2}\right )} - \frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 31, normalized size = 1.07 \begin {gather*} -\frac {4}{x}-\frac {x^2}{2}-\frac {x\,{\mathrm {e}}^{-{\mathrm {e}}^2}\,\left ({\mathrm {e}}^{{\mathrm {e}}^2+1}+5\,{\mathrm {e}}^{{\mathrm {e}}^2}-4\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 42, normalized size = 1.45 \begin {gather*} \frac {- x^{2} e^{e^{2}} - x \left (-4 + e e^{e^{2}} + 5 e^{e^{2}}\right ) - \frac {8 e^{e^{2}}}{x}}{2 e^{e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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