Optimal. Leaf size=27 \[ \left (40 e^{-x \left (3+4 x^2\right )^2} x-x (3+x)\right ) \log (x) \]
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Rubi [B] time = 3.29, antiderivative size = 69, normalized size of antiderivative = 2.56, number of steps used = 5, number of rules used = 5, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6741, 6742, 6688, 2288, 2554} \begin {gather*} x^2 (-\log (x))+\frac {40 e^{-x \left (4 x^2+3\right )^2} \left (80 x^5+72 x^3+9 x\right ) \log (x)}{16 \left (4 x^2+3\right ) x^2+\left (4 x^2+3\right )^2}-3 x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2288
Rule 2554
Rule 6688
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-x \left (3+4 x^2\right )^2} \left (40+e^{9 x+24 x^3+16 x^5} (-3-x)+\left (40+e^{9 x+24 x^3+16 x^5} (-3-2 x)-360 x-2880 x^3-3200 x^5\right ) \log (x)\right ) \, dx\\ &=\int \left (-3+40 e^{-x \left (3+4 x^2\right )^2}-x-e^{-x \left (3+4 x^2\right )^2} \left (-40+3 e^{x \left (3+4 x^2\right )^2}+360 x+2 e^{x \left (3+4 x^2\right )^2} x+2880 x^3+3200 x^5\right ) \log (x)\right ) \, dx\\ &=-3 x-\frac {x^2}{2}+40 \int e^{-x \left (3+4 x^2\right )^2} \, dx-\int e^{-x \left (3+4 x^2\right )^2} \left (-40+3 e^{x \left (3+4 x^2\right )^2}+360 x+2 e^{x \left (3+4 x^2\right )^2} x+2880 x^3+3200 x^5\right ) \log (x) \, dx\\ &=-3 x-\frac {x^2}{2}-3 x \log (x)-x^2 \log (x)+\frac {40 e^{-x \left (3+4 x^2\right )^2} \left (9 x+72 x^3+80 x^5\right ) \log (x)}{16 x^2 \left (3+4 x^2\right )+\left (3+4 x^2\right )^2}+40 \int e^{-x \left (3+4 x^2\right )^2} \, dx+\int \left (3-40 e^{-x \left (3+4 x^2\right )^2}+x\right ) \, dx\\ &=-3 x \log (x)-x^2 \log (x)+\frac {40 e^{-x \left (3+4 x^2\right )^2} \left (9 x+72 x^3+80 x^5\right ) \log (x)}{16 x^2 \left (3+4 x^2\right )+\left (3+4 x^2\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.35, size = 25, normalized size = 0.93 \begin {gather*} \left (-3+40 e^{-x \left (3+4 x^2\right )^2}-x\right ) x \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 46, normalized size = 1.70 \begin {gather*} -{\left ({\left (x^{2} + 3 \, x\right )} e^{\left (16 \, x^{5} + 24 \, x^{3} + 9 \, x\right )} - 40 \, x\right )} e^{\left (-16 \, x^{5} - 24 \, x^{3} - 9 \, x\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 33, normalized size = 1.22 \begin {gather*} -x^{2} \log \relax (x) + 40 \, x e^{\left (-16 \, x^{5} - 24 \, x^{3} - 9 \, x\right )} \log \relax (x) - 3 \, x \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 49, normalized size = 1.81
method | result | size |
risch | \(-x \left ({\mathrm e}^{\left (4 x^{2}+3\right )^{2} x} x +3 \,{\mathrm e}^{\left (4 x^{2}+3\right )^{2} x}-40\right ) {\mathrm e}^{-\left (4 x^{2}+3\right )^{2} x} \ln \relax (x )\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 62, normalized size = 2.30 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (80 \, x e^{\left (-16 \, x^{5}\right )} \log \relax (x) + {\left (x^{2} - 2 \, {\left (x^{2} + 3 \, x\right )} \log \relax (x) + 6 \, x\right )} e^{\left (24 \, x^{3} + 9 \, x\right )}\right )} e^{\left (-24 \, x^{3} - 9 \, x\right )} - 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -{\mathrm {e}}^{-16\,x^5-24\,x^3-9\,x}\,\left ({\mathrm {e}}^{16\,x^5+24\,x^3+9\,x}\,\left (x+3\right )+\ln \relax (x)\,\left (360\,x+{\mathrm {e}}^{16\,x^5+24\,x^3+9\,x}\,\left (2\,x+3\right )+2880\,x^3+3200\,x^5-40\right )-40\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 34, normalized size = 1.26 \begin {gather*} 40 x e^{- 16 x^{5} - 24 x^{3} - 9 x} \log {\relax (x )} + \left (- x^{2} - 3 x\right ) \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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