Optimal. Leaf size=36 \[ 5+\frac {\left (-1+\frac {5}{x}\right )^2 x}{1-e^{e^{(-x+x \log (2))^2}}-2 x} \]
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Rubi [F]
time = 14.84, 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 {-25+100 x-19 x^2+e^{e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} \left (25-x^2+e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)} \left (50 x^2-20 x^3+2 x^4+\left (-100 x^2+40 x^3-4 x^4\right ) \log (2)+\left (50 x^2-20 x^3+2 x^4\right ) \log ^2(2)\right )\right )}{x^2+e^{2 e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} x^2-4 x^3+4 x^4+e^{e^{x^2-2 x^2 \log (2)+x^2 \log ^2(2)}} \left (-2 x^2+4 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4^{-x^2} (5-x) \left (4^{x^2} e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}} (5+x)+4^{x^2} (-5+19 x)-2 \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) (-5+x) x^2 (-1+\log (2))^2\right )}{\left (1-e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}-2 x\right )^2 x^2} \, dx\\ &=\int \left (-\frac {(-5+x) \left (-5+5 e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+19 x+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}} x\right )}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) (-5+x)^2 (-1+\log (2))^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}\right ) \, dx\\ &=(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) (-5+x)^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \frac {(-5+x) \left (-5+5 e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+19 x+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}} x\right )}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx\\ &=(-1+\log (2))^2 \int \left (\frac {25\ 2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}-\frac {5\ 2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}\right ) \, dx-\int \left (-\frac {2 (-5+x)^2}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {(-5+x) (5+x)}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )}\right ) \, dx\\ &=2 \int \frac {(-5+x)^2}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\left (5 (1-\log (2))^2\right ) \int \frac {2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+\left (25 (1-\log (2))^2\right ) \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \frac {(-5+x) (5+x)}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )} \, dx\\ &=2 \int \left (-\frac {10}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {25}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}+\frac {x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2}\right ) \, dx-\left (5 (1-\log (2))^2\right ) \int \frac {2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+\left (25 (1-\log (2))^2\right ) \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \left (\frac {1}{-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x}-\frac {25}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )}\right ) \, dx\\ &=2 \int \frac {x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-20 \int \frac {1}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+25 \int \frac {1}{x^2 \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )} \, dx+50 \int \frac {1}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\left (5 (1-\log (2))^2\right ) \int \frac {2^{2-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+\left (25 (1-\log (2))^2\right ) \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right )}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx+(-1+\log (2))^2 \int \frac {2^{1-2 x^2} \exp \left (4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}+x^2 \left (1+\log ^2(2)\right )\right ) x^2}{\left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )^2} \, dx-\int \frac {1}{-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.55, size = 39, normalized size = 1.08 \begin {gather*} -\frac {(-5+x)^2}{x \left (-1+e^{4^{-x^2} e^{x^2 \left (1+\log ^2(2)\right )}}+2 x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 39, normalized size = 1.08
method | result | size |
risch | \(-\frac {x^{2}-10 x +25}{x \left (-1+{\mathrm e}^{\left (\frac {1}{4}\right )^{x^{2}} {\mathrm e}^{x^{2} \left (\ln \left (2\right )^{2}+1\right )}}+2 x \right )}\) | \(39\) |
norman | \(\frac {-25+\frac {19 x}{2}+\frac {{\mathrm e}^{{\mathrm e}^{x^{2} \ln \left (2\right )^{2}-2 x^{2} \ln \left (2\right )+x^{2}}} x}{2}}{x \left (-1+{\mathrm e}^{{\mathrm e}^{x^{2} \ln \left (2\right )^{2}-2 x^{2} \ln \left (2\right )+x^{2}}}+2 x \right )}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 44, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 10 \, x + 25}{2 \, x^{2} + x e^{\left (e^{\left (x^{2} \log \left (2\right )^{2} - 2 \, x^{2} \log \left (2\right ) + x^{2}\right )}\right )} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 44, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 10 \, x + 25}{2 \, x^{2} + x e^{\left (e^{\left (x^{2} \log \left (2\right )^{2} - 2 \, x^{2} \log \left (2\right ) + x^{2}\right )}\right )} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 39, normalized size = 1.08 \begin {gather*} \frac {- x^{2} + 10 x - 25}{2 x^{2} + x e^{e^{- 2 x^{2} \log {\left (2 \right )} + x^{2} \log {\left (2 \right )}^{2} + x^{2}}} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 44, normalized size = 1.22 \begin {gather*} -\frac {x^{2} - 10 \, x + 25}{2 \, x^{2} + x e^{\left (e^{\left (x^{2} \log \left (2\right )^{2} - 2 \, x^{2} \log \left (2\right ) + x^{2}\right )}\right )} - 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 = -0.03 \begin {gather*} \int \frac {100\,x+{\mathrm {e}}^{{\mathrm {e}}^{x^2\,{\ln \left (2\right )}^2-2\,x^2\,\ln \left (2\right )+x^2}}\,\left ({\mathrm {e}}^{x^2\,{\ln \left (2\right )}^2-2\,x^2\,\ln \left (2\right )+x^2}\,\left ({\ln \left (2\right )}^2\,\left (2\,x^4-20\,x^3+50\,x^2\right )-\ln \left (2\right )\,\left (4\,x^4-40\,x^3+100\,x^2\right )+50\,x^2-20\,x^3+2\,x^4\right )-x^2+25\right )-19\,x^2-25}{x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2\,{\ln \left (2\right )}^2-2\,x^2\,\ln \left (2\right )+x^2}}-{\mathrm {e}}^{{\mathrm {e}}^{x^2\,{\ln \left (2\right )}^2-2\,x^2\,\ln \left (2\right )+x^2}}\,\left (2\,x^2-4\,x^3\right )+x^2-4\,x^3+4\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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