Optimal. Leaf size=28 \[ 4+2 e^{-\log ^2(x)}-\log \left (5+\left (e^{4+x^2}+x\right )^2\right ) \]
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Rubi [A] time = 0.64, antiderivative size = 38, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 5, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6688, 6684, 2276, 2205, 2209} \begin {gather*} 2 e^{-\log ^2(x)}-\log \left (x^2+2 e^{x^2+4} x+e^{2 x^2+8}+5\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2205
Rule 2209
Rule 2276
Rule 6684
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (x+2 e^{8+2 x^2} x+e^{4+x^2} \left (1+2 x^2\right )\right )}{5+e^{8+2 x^2}+2 e^{4+x^2} x+x^2}-\frac {4 e^{-\log ^2(x)} \log (x)}{x}\right ) \, dx\\ &=-\left (2 \int \frac {x+2 e^{8+2 x^2} x+e^{4+x^2} \left (1+2 x^2\right )}{5+e^{8+2 x^2}+2 e^{4+x^2} x+x^2} \, dx\right )-4 \int \frac {e^{-\log ^2(x)} \log (x)}{x} \, dx\\ &=-\log \left (5+e^{8+2 x^2}+2 e^{4+x^2} x+x^2\right )-4 \operatorname {Subst}\left (\int e^{-x^2} x \, dx,x,\log (x)\right )\\ &=2 e^{-\log ^2(x)}-\log \left (5+e^{8+2 x^2}+2 e^{4+x^2} x+x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.34, size = 38, normalized size = 1.36 \begin {gather*} 2 e^{-\log ^2(x)}-\log \left (5+e^{8+2 x^2}+2 e^{4+x^2} x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 40, normalized size = 1.43 \begin {gather*} -{\left (e^{\left (\log \relax (x)^{2}\right )} \log \left (x^{2} + 2 \, x e^{\left (x^{2} + 4\right )} + e^{\left (2 \, x^{2} + 8\right )} + 5\right ) - 2\right )} e^{\left (-\log \relax (x)^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 40, normalized size = 1.43 \begin {gather*} -{\left (e^{\left (\log \relax (x)^{2}\right )} \log \left (x^{2} + 2 \, x e^{\left (x^{2} + 4\right )} + e^{\left (2 \, x^{2} + 8\right )} + 5\right ) - 2\right )} e^{\left (-\log \relax (x)^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 37, normalized size = 1.32
method | result | size |
risch | \(8-\ln \left (2 x \,{\mathrm e}^{x^{2}+4}+x^{2}+{\mathrm e}^{2 x^{2}+8}+5\right )+2 \,{\mathrm e}^{-\ln \relax (x )^{2}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 38, normalized size = 1.36 \begin {gather*} 2 \, e^{\left (-\log \relax (x)^{2}\right )} - \log \left ({\left (x^{2} + 2 \, x e^{\left (x^{2} + 4\right )} + e^{\left (2 \, x^{2} + 8\right )} + 5\right )} e^{\left (-8\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.74, size = 36, normalized size = 1.29 \begin {gather*} 2\,{\mathrm {e}}^{-{\ln \relax (x)}^2}-\ln \left ({\mathrm {e}}^8\,{\mathrm {e}}^{2\,x^2}+x^2+2\,x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^4+5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 32, normalized size = 1.14 \begin {gather*} - \log {\left (x^{2} + 2 x e^{x^{2} + 4} + e^{2 x^{2} + 8} + 5 \right )} + 2 e^{- \log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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