Optimal. Leaf size=22 \[ 2 x+\frac {5 e^{-2 x^2} (-8+3 e x)}{x} \]
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Rubi [A]
time = 0.35, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps
used = 5, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6873, 12, 6874,
2326} \begin {gather*} 2 x-\frac {5 e^{-2 x^2} \left (8 x^2-3 e x^3\right )}{x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-2 x^2} \left (20+80 x^2+e^{2 x^2} x^2-30 e x^3\right )}{x^2} \, dx\\ &=2 \int \frac {e^{-2 x^2} \left (20+80 x^2+e^{2 x^2} x^2-30 e x^3\right )}{x^2} \, dx\\ &=2 \int \left (1-\frac {10 e^{-2 x^2} \left (-2-8 x^2+3 e x^3\right )}{x^2}\right ) \, dx\\ &=2 x-20 \int \frac {e^{-2 x^2} \left (-2-8 x^2+3 e x^3\right )}{x^2} \, dx\\ &=2 x-\frac {5 e^{-2 x^2} \left (8 x^2-3 e x^3\right )}{x^3}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.75, size = 27, normalized size = 1.23 \begin {gather*} 15 e^{1-2 x^2}-\frac {40 e^{-2 x^2}}{x}+2 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 26, normalized size = 1.18
method | result | size |
risch | \(\frac {5 \left (3 x \,{\mathrm e}-8\right ) {\mathrm e}^{-2 x^{2}}}{x}+2 x\) | \(23\) |
default | \(2 x -\frac {40 \,{\mathrm e}^{-2 x^{2}}}{x}+15 \,{\mathrm e} \,{\mathrm e}^{-2 x^{2}}\) | \(26\) |
norman | \(\frac {\left (-40+15 x \,{\mathrm e}+2 x^{2} {\mathrm e}^{2 x^{2}}\right ) {\mathrm e}^{-2 x^{2}}}{x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.49, size = 48, normalized size = 2.18 \begin {gather*} 40 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x\right ) + 2 \, x - \frac {20 \, \sqrt {2} \sqrt {x^{2}} \Gamma \left (-\frac {1}{2}, 2 \, x^{2}\right )}{x} + 15 \, e^{\left (-2 \, x^{2} + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 28, normalized size = 1.27 \begin {gather*} \frac {{\left (2 \, x^{2} e^{\left (2 \, x^{2}\right )} + 15 \, x e - 40\right )} e^{\left (-2 \, x^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.86 \begin {gather*} 2 x + \frac {\left (15 e x - 40\right ) e^{- 2 x^{2}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 29, normalized size = 1.32 \begin {gather*} \frac {2 \, x^{2} + 15 \, x e^{\left (-2 \, x^{2} + 1\right )} - 40 \, e^{\left (-2 \, x^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 25, normalized size = 1.14 \begin {gather*} 2\,x+15\,\mathrm {e}\,{\mathrm {e}}^{-2\,x^2}-\frac {40\,{\mathrm {e}}^{-2\,x^2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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