Optimal. Leaf size=29 \[ \frac {e^{-3+e^x} x}{2 \left (e^{2+4 x}-\frac {\log (2)}{x}\right )} \]
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Rubi [F]
time = 3.99, 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 \frac {e^{-3+e^x} \left (e^{2+4 x} \left (x^2-4 x^3+e^x x^3\right )-2 x \log (2)-e^x x^2 \log (2)\right )}{2 e^{4+8 x} x^2-4 e^{2+4 x} x \log (2)+2 \log ^2(2)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-3+e^x} \left (e^{2+4 x} \left (x^2-4 x^3+e^x x^3\right )-2 x \log (2)-e^x x^2 \log (2)\right )}{2 \left (e^{2+4 x} x-\log (2)\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{-3+e^x} \left (e^{2+4 x} \left (x^2-4 x^3+e^x x^3\right )-2 x \log (2)-e^x x^2 \log (2)\right )}{\left (e^{2+4 x} x-\log (2)\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {e^{-3+e^x} x \left (1-4 x+e^x x\right )}{e^{2+4 x} x-\log (2)}-\frac {e^{-3+e^x} x (1+4 x) \log (2)}{\left (e^{2+4 x} x-\log (2)\right )^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{-3+e^x} x \left (1-4 x+e^x x\right )}{e^{2+4 x} x-\log (2)} \, dx-\frac {1}{2} \log (2) \int \frac {e^{-3+e^x} x (1+4 x)}{\left (e^{2+4 x} x-\log (2)\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {e^{-3+e^x} x}{e^{2+4 x} x-\log (2)}-\frac {4 e^{-3+e^x} x^2}{e^{2+4 x} x-\log (2)}+\frac {e^{-3+e^x+x} x^2}{e^{2+4 x} x-\log (2)}\right ) \, dx-\frac {1}{2} \log (2) \int \left (\frac {e^{-3+e^x} x}{\left (e^{2+4 x} x-\log (2)\right )^2}+\frac {4 e^{-3+e^x} x^2}{\left (e^{2+4 x} x-\log (2)\right )^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{-3+e^x} x}{e^{2+4 x} x-\log (2)} \, dx+\frac {1}{2} \int \frac {e^{-3+e^x+x} x^2}{e^{2+4 x} x-\log (2)} \, dx-2 \int \frac {e^{-3+e^x} x^2}{e^{2+4 x} x-\log (2)} \, dx-\frac {1}{2} \log (2) \int \frac {e^{-3+e^x} x}{\left (e^{2+4 x} x-\log (2)\right )^2} \, dx-(2 \log (2)) \int \frac {e^{-3+e^x} x^2}{\left (e^{2+4 x} x-\log (2)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 30, normalized size = 1.03 \begin {gather*} \frac {e^{-3+e^x} x^2}{2 \left (e^{2+4 x} x-\log (2)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 26, normalized size = 0.90
method | result | size |
risch | \(\frac {x^{2} {\mathrm e}^{{\mathrm e}^{x}-3}}{2 x \,{\mathrm e}^{4 x +2}-2 \ln \left (2\right )}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 25, normalized size = 0.86 \begin {gather*} \frac {x^{2} e^{\left (e^{x}\right )}}{2 \, {\left (x e^{\left (4 \, x + 5\right )} - e^{3} \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 25, normalized size = 0.86 \begin {gather*} \frac {x^{2} e^{\left (e^{x} - 3\right )}}{2 \, {\left (x e^{\left (4 \, x + 2\right )} - \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 26, normalized size = 0.90 \begin {gather*} \frac {x^{2} e^{e^{x} - 3}}{2 x e^{2} e^{4 x} - 2 \log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 29, normalized size = 1.00 \begin {gather*} \frac {x^{2} e^{\left (x + e^{x}\right )}}{2 \, {\left (x e^{\left (5 \, x + 5\right )} - e^{\left (x + 3\right )} \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.50, size = 25, normalized size = 0.86 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-5}}{2\,\left (x\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{-2}\,\ln \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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