Optimal. Leaf size=32 \[ e^{e^{-2 x-\frac {2+x-\log (4)}{x}} x} x \left (1+e^{e^4}+x\right ) \]
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Rubi [F]
time = 16.67, 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 \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \left (2+3 x-x^2-2 x^3+e^{\frac {2+x+2 x^2-\log (4)}{x}} (1+2 x)+e^{e^4} \left (2+e^{\frac {2+x+2 x^2-\log (4)}{x}}+x-2 x^2-\log (4)\right )+(-1-x) \log (4)\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right )+3 \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x-\exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^2-2 \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^3+4^{-1/x} \exp \left (1+\frac {2}{x}+2 x+e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) (1+2 x)+2^{-2/x} \exp \left (e^4+e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \left (e^{1+\frac {2}{x}+2 x}+2^{2/x} x-2^{1+\frac {2}{x}} x^2+2^{1+\frac {2}{x}} (1-\log (2))\right )-\exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) (1+x) \log (4)\right ) \, dx\\ &=2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx-2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^3 \, dx+3 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) (1+x) \, dx-\int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^2 \, dx+\int 4^{-1/x} \exp \left (1+\frac {2}{x}+2 x+e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) (1+2 x) \, dx+\int 2^{-2/x} \exp \left (e^4+e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \left (e^{1+\frac {2}{x}+2 x}+2^{2/x} x-2^{1+\frac {2}{x}} x^2+2^{1+\frac {2}{x}} (1-\log (2))\right ) \, dx\\ &=2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx-2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^3 \, dx+3 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx-\log (4) \int \left (\exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right )+\exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x\right ) \, dx-\int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^2 \, dx+\int e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} (1+2 x) \, dx+\int e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \left (e^{1+\frac {2}{x}+2 x}+4^{\frac {1}{x}} \left (2+x-2 x^2-\log (4)\right )\right ) \, dx\\ &=2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx-2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^3 \, dx+3 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx-\int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^2 \, dx+\int \left (e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x}+2 e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x\right ) \, dx+\int \left (e^{e^4+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x}-4^{\frac {1}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \left (-2-x+2 x^2+\log (4)\right )\right ) \, dx\\ &=2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx+2 \int e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x \, dx-2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^3 \, dx+3 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx+\int e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \, dx+\int e^{e^4+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \, dx-\int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^2 \, dx-\int 4^{\frac {1}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \left (-2-x+2 x^2+\log (4)\right ) \, dx\\ &=2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx+2 \int e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x \, dx-2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^3 \, dx+3 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx+\int e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \, dx+\int e^{e^4+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \, dx-\int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^2 \, dx-\int \left (-4^{\frac {1}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x+2^{1+\frac {2}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x^2-2^{1+\frac {2}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} (1-\log (2))\right ) \, dx\\ &=2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx+2 \int e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x \, dx-2 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^3 \, dx+3 \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx-(-1+\log (2)) \int 2^{1+\frac {2}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) \, dx-\log (4) \int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x \, dx+\int e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \, dx+\int e^{e^4+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} \, dx+\int 4^{\frac {1}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x \, dx-\int 2^{1+\frac {2}{x}} e^{-1+e^4-\frac {2}{x}-2 x+4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x^2 \, dx-\int \exp \left (e^{-\frac {2+x+2 x^2-\log (4)}{x}} x-\frac {2+x+2 x^2-\log (4)}{x}\right ) x^2 \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 31, normalized size = 0.97 \begin {gather*} e^{4^{\frac {1}{x}} e^{-1-\frac {2}{x}-2 x} x} x \left (1+e^{e^4}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 31, normalized size = 0.97
method | result | size |
risch | \(x \left ({\mathrm e}^{{\mathrm e}^{4}}+x +1\right ) {\mathrm e}^{x \,{\mathrm e}^{\frac {-2 x^{2}+2 \ln \left (2\right )-x -2}{x}}}\) | \(31\) |
norman | \(\left (x^{2} {\mathrm e}^{\frac {-2 \ln \left (2\right )+2 x^{2}+x +2}{x}} {\mathrm e}^{x \,{\mathrm e}^{-\frac {-2 \ln \left (2\right )+2 x^{2}+x +2}{x}}}+\left ({\mathrm e}^{{\mathrm e}^{4}}+1\right ) x \,{\mathrm e}^{\frac {-2 \ln \left (2\right )+2 x^{2}+x +2}{x}} {\mathrm e}^{x \,{\mathrm e}^{-\frac {-2 \ln \left (2\right )+2 x^{2}+x +2}{x}}}\right ) {\mathrm e}^{-\frac {-2 \ln \left (2\right )+2 x^{2}+x +2}{x}}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 33, normalized size = 1.03 \begin {gather*} {\left (x^{2} + x {\left (e^{\left (e^{4}\right )} + 1\right )}\right )} e^{\left (x e^{\left (-2 \, x + \frac {2 \, \log \left (2\right )}{x} - \frac {2}{x} - 1\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (28) = 56\).
time = 0.35, size = 69, normalized size = 2.16 \begin {gather*} {\left (x^{2} + x e^{\left (e^{4}\right )} + x\right )} e^{\left (\frac {x^{2} e^{\left (-\frac {2 \, x^{2} + x - 2 \, \log \left (2\right ) + 2}{x}\right )} - 2 \, x^{2} - x + 2 \, \log \left (2\right ) - 2}{x} + \frac {2 \, x^{2} + x - 2 \, \log \left (2\right ) + 2}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.89, size = 31, normalized size = 0.97 \begin {gather*} \left (x^{2} + x + x e^{e^{4}}\right ) e^{x e^{- \frac {2 x^{2} + x - 2 \log {\left (2 \right )} + 2}{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*} \text {could not integrate} \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 = -0.03 \begin {gather*} \int {\mathrm {e}}^{-\frac {2\,x^2+x-2\,\ln \left (2\right )+2}{x}}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {2\,x^2+x-2\,\ln \left (2\right )+2}{x}}}\,\left (3\,x+{\mathrm {e}}^{\frac {2\,x^2+x-2\,\ln \left (2\right )+2}{x}}\,\left (2\,x+1\right )+{\mathrm {e}}^{{\mathrm {e}}^4}\,\left (x-2\,\ln \left (2\right )+{\mathrm {e}}^{\frac {2\,x^2+x-2\,\ln \left (2\right )+2}{x}}-2\,x^2+2\right )-2\,\ln \left (2\right )\,\left (x+1\right )-x^2-2\,x^3+2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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