Optimal. Leaf size=22 \[ -x+e^{-2 x^2} \left (\frac {2}{x}+x \log (8)\right ) \]
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Rubi [C] time = 0.57, antiderivative size = 94, normalized size of antiderivative = 4.27, number of steps used = 9, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6742, 2214, 2205, 2212} \begin {gather*} 2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} x\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \log (8) \text {erf}\left (\sqrt {2} x\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} (8-\log (8)) \text {erf}\left (\sqrt {2} x\right )+\frac {2 e^{-2 x^2}}{x}+e^{-2 x^2} x \log (8)-x \end {gather*}
Antiderivative was successfully verified.
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Rule 2205
Rule 2212
Rule 2214
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {e^{-2 x^2} \left (-2-x^2 (8-\log (8))-4 x^4 \log (8)\right )}{x^2}\right ) \, dx\\ &=-x+\int \frac {e^{-2 x^2} \left (-2-x^2 (8-\log (8))-4 x^4 \log (8)\right )}{x^2} \, dx\\ &=-x+\int \left (-\frac {2 e^{-2 x^2}}{x^2}-8 e^{-2 x^2} \left (1-\frac {3 \log (2)}{8}\right )-4 e^{-2 x^2} x^2 \log (8)\right ) \, dx\\ &=-x-2 \int \frac {e^{-2 x^2}}{x^2} \, dx-(8-\log (8)) \int e^{-2 x^2} \, dx-(4 \log (8)) \int e^{-2 x^2} x^2 \, dx\\ &=\frac {2 e^{-2 x^2}}{x}-x-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} x\right ) (8-\log (8))+e^{-2 x^2} x \log (8)+8 \int e^{-2 x^2} \, dx-\log (8) \int e^{-2 x^2} \, dx\\ &=\frac {2 e^{-2 x^2}}{x}-x+2 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} x\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} x\right ) (8-\log (8))+e^{-2 x^2} x \log (8)-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} x\right ) \log (8)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 22, normalized size = 1.00 \begin {gather*} -x+e^{-2 x^2} \left (\frac {2}{x}+x \log (8)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 30, normalized size = 1.36 \begin {gather*} -\frac {{\left (x^{2} e^{\left (2 \, x^{2}\right )} - 3 \, x^{2} \log \relax (2) - 2\right )} e^{\left (-2 \, x^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 31, normalized size = 1.41 \begin {gather*} \frac {3 \, x^{2} e^{\left (-2 \, x^{2}\right )} \log \relax (2) - x^{2} + 2 \, e^{\left (-2 \, x^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 24, normalized size = 1.09
method | result | size |
risch | \(-x +\frac {\left (3 x^{2} \ln \relax (2)+2\right ) {\mathrm e}^{-2 x^{2}}}{x}\) | \(24\) |
default | \(-x +\frac {2 \,{\mathrm e}^{-2 x^{2}}}{x}+3 \ln \relax (2) x \,{\mathrm e}^{-2 x^{2}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.49, size = 80, normalized size = 3.64 \begin {gather*} \frac {3}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x\right ) \log \relax (2) - 2 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x\right ) - \frac {3}{4} \, {\left (\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x\right ) - 4 \, x e^{\left (-2 \, x^{2}\right )}\right )} \log \relax (2) - x + \frac {\sqrt {2} \sqrt {x^{2}} \Gamma \left (-\frac {1}{2}, 2 \, x^{2}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 26, normalized size = 1.18 \begin {gather*} x\,\left (3\,{\mathrm {e}}^{-2\,x^2}\,\ln \relax (2)-1\right )+\frac {2\,{\mathrm {e}}^{-2\,x^2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 0.86 \begin {gather*} - x + \frac {\left (3 x^{2} \log {\relax (2 )} + 2\right ) e^{- 2 x^{2}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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