Optimal. Leaf size=25 \[ x^2 \left (e^{2+4 x+e^{-2 x} x^2}-\log (x)\right ) \]
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Rubi [F] time = 1.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{-2 x} \left (-e^{2 x} x+e^{e^{-2 x} \left (x^2+e^{2 x} (2+4 x)\right )} \left (2 x^3-2 x^4+e^{2 x} \left (2 x+4 x^2\right )\right )-2 e^{2 x} x \log (x)\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int x \left (-1+2 e^{2+2 x+e^{-2 x} x^2} \left (-\left ((-1+x) x^2\right )+e^{2 x} (1+2 x)\right )-2 \log (x)\right ) \, dx\\ &=\int \left (2 e^{2+2 x+e^{-2 x} x^2} x \left (e^{2 x}+2 e^{2 x} x+x^2-x^3\right )-x (1+2 \log (x))\right ) \, dx\\ &=2 \int e^{2+2 x+e^{-2 x} x^2} x \left (e^{2 x}+2 e^{2 x} x+x^2-x^3\right ) \, dx-\int x (1+2 \log (x)) \, dx\\ &=-x^2 \log (x)+2 \int \left (-e^{2+2 x+e^{-2 x} x^2} (-1+x) x^3+e^{2+4 x+e^{-2 x} x^2} x (1+2 x)\right ) \, dx\\ &=-x^2 \log (x)-2 \int e^{2+2 x+e^{-2 x} x^2} (-1+x) x^3 \, dx+2 \int e^{2+4 x+e^{-2 x} x^2} x (1+2 x) \, dx\\ &=-x^2 \log (x)+2 \int \left (e^{2+4 x+e^{-2 x} x^2} x+2 e^{2+4 x+e^{-2 x} x^2} x^2\right ) \, dx-2 \int \left (-e^{2+2 x+e^{-2 x} x^2} x^3+e^{2+2 x+e^{-2 x} x^2} x^4\right ) \, dx\\ &=-x^2 \log (x)+2 \int e^{2+4 x+e^{-2 x} x^2} x \, dx+2 \int e^{2+2 x+e^{-2 x} x^2} x^3 \, dx-2 \int e^{2+2 x+e^{-2 x} x^2} x^4 \, dx+4 \int e^{2+4 x+e^{-2 x} x^2} x^2 \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.44, size = 25, normalized size = 1.00 \begin {gather*} x^2 \left (e^{2+4 x+e^{-2 x} x^2}-\log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 33, normalized size = 1.32 \begin {gather*} x^{2} e^{\left ({\left (x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}\right )} - x^{2} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left (2 \, x e^{\left (2 \, x\right )} \log \relax (x) + 2 \, {\left (x^{4} - x^{3} - {\left (2 \, x^{2} + x\right )} e^{\left (2 \, x\right )}\right )} e^{\left ({\left (x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}\right )} + x e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 36, normalized size = 1.44
method | result | size |
risch | \(-x^{2} \ln \relax (x )+{\mathrm e}^{\left (4 x \,{\mathrm e}^{2 x}+x^{2}+2 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}} x^{2}\) | \(36\) |
default | \(-x^{2} \ln \relax (x )+{\mathrm e}^{\left (4 x \,{\mathrm e}^{2 x}+x^{2}+2 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}} x^{2}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x^{2} \log \relax (x) - \int -2 \, {\left ({\left (2 \, x^{2} e^{2} + x e^{2}\right )} e^{\left (4 \, x\right )} - {\left (x^{4} e^{2} - x^{3} e^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (x^{2} e^{\left (-2 \, x\right )}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 27, normalized size = 1.08 \begin {gather*} x^2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-2\,x}}-x^2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 29, normalized size = 1.16 \begin {gather*} x^{2} e^{\left (x^{2} + \left (4 x + 2\right ) e^{2 x}\right ) e^{- 2 x}} - x^{2} \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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