Optimal. Leaf size=20 \[ 5+e^{-3-x^2}-\log \left (\log \left (2+x^2\right )\right ) \]
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Rubi [A] time = 0.33, antiderivative size = 19, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6688, 2209, 2475, 2390, 2302, 29} \begin {gather*} e^{-x^2-3}-\log \left (\log \left (x^2+2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2209
Rule 2302
Rule 2390
Rule 2475
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2 e^{-3-x^2} x-\frac {2 x}{\left (2+x^2\right ) \log \left (2+x^2\right )}\right ) \, dx\\ &=-\left (2 \int e^{-3-x^2} x \, dx\right )-2 \int \frac {x}{\left (2+x^2\right ) \log \left (2+x^2\right )} \, dx\\ &=e^{-3-x^2}-\operatorname {Subst}\left (\int \frac {1}{(2+x) \log (2+x)} \, dx,x,x^2\right )\\ &=e^{-3-x^2}-\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,2+x^2\right )\\ &=e^{-3-x^2}-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (2+x^2\right )\right )\\ &=e^{-3-x^2}-\log \left (\log \left (2+x^2\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 19, normalized size = 0.95 \begin {gather*} e^{-3-x^2}-\log \left (\log \left (2+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} - 3\right )} - \log \left (\log \left (x^{2} + 2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} - 3\right )} - \log \left (\log \left (x^{2} + 2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 19, normalized size = 0.95
method | result | size |
default | \({\mathrm e}^{-x^{2}-3}-\ln \left (\ln \left (x^{2}+2\right )\right )\) | \(19\) |
norman | \({\mathrm e}^{-x^{2}-3}-\ln \left (\ln \left (x^{2}+2\right )\right )\) | \(19\) |
risch | \({\mathrm e}^{-x^{2}-3}-\ln \left (\ln \left (x^{2}+2\right )\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} - 3\right )} - \log \left (\log \left (x^{2} + 2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.68, size = 18, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{-x^2-3}-\ln \left (\ln \left (x^2+2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 15, normalized size = 0.75 \begin {gather*} e^{- x^{2} - 3} - \log {\left (\log {\left (x^{2} + 2 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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