Optimal. Leaf size=31 \[ e^{x^2}+3 e^{-x^2 \left (4-x-\frac {x^4}{\log ^2(2)}\right )^2} \]
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Rubi [A] time = 17.53, antiderivative size = 37, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 6, integrand size = 150, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {12, 6741, 6742, 2225, 2209, 6706} \begin {gather*} 3 \exp \left (-\frac {x^2 \left (x^4+x \log ^2(2)-4 \log ^2(2)\right )^2}{\log ^4(2)}\right )+e^{x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2225
Rule 6706
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \exp \left (-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}\right ) \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 \exp \left (x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}\right ) x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right ) \, dx}{\log ^4(2)}\\ &=\frac {\int 2 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (-15 x^8+72 x^4 \log ^2(2)-21 x^5 \log ^2(2)-48 \log ^4(2)+\exp \left (x^2 \left (17-8 x+x^2+\frac {x^8}{\log ^4(2)}-\frac {8 x^4}{\log ^2(2)}+\frac {2 x^5}{\log ^2(2)}\right )\right ) \log ^4(2)+36 x \log ^4(2)-6 x^2 \log ^4(2)\right ) \, dx}{\log ^4(2)}\\ &=\frac {2 \int \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (-15 x^8+72 x^4 \log ^2(2)-21 x^5 \log ^2(2)-48 \log ^4(2)+\exp \left (x^2 \left (17-8 x+x^2+\frac {x^8}{\log ^4(2)}-\frac {8 x^4}{\log ^2(2)}+\frac {2 x^5}{\log ^2(2)}\right )\right ) \log ^4(2)+36 x \log ^4(2)-6 x^2 \log ^4(2)\right ) \, dx}{\log ^4(2)}\\ &=\frac {2 \int \left (\exp \left (17 x^2-8 x^3+x^4+\frac {x^{10}}{\log ^4(2)}-\frac {8 x^6}{\log ^2(2)}+\frac {2 x^7}{\log ^2(2)}-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \log ^4(2)-3 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (5 x^8-24 x^4 \log ^2(2)+7 x^5 \log ^2(2)+16 \log ^4(2)-12 x \log ^4(2)+2 x^2 \log ^4(2)\right )\right ) \, dx}{\log ^4(2)}\\ &=2 \int \exp \left (17 x^2-8 x^3+x^4+\frac {x^{10}}{\log ^4(2)}-\frac {8 x^6}{\log ^2(2)}+\frac {2 x^7}{\log ^2(2)}-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \, dx-\frac {6 \int \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (5 x^8-24 x^4 \log ^2(2)+7 x^5 \log ^2(2)+16 \log ^4(2)-12 x \log ^4(2)+2 x^2 \log ^4(2)\right ) \, dx}{\log ^4(2)}\\ &=3 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right )+2 \int e^{x^2} x \, dx\\ &=e^{x^2}+3 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.25, size = 56, normalized size = 1.81 \begin {gather*} e^{-16 x^2} \left (e^{17 x^2}+3 e^{x^3 \left (8-x-\frac {x^7}{\log ^4(2)}+\frac {8 x^3}{\log ^2(2)}-\frac {2 x^4}{\log ^2(2)}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 137, normalized size = 4.42 \begin {gather*} e^{\left (\frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 17 \, x^{2}\right )} \log \relax (2)^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \relax (2)^{2}}{\log \relax (2)^{4}} - \frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \relax (2)^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \relax (2)^{2}}{\log \relax (2)^{4}}\right )} + 3 \, e^{\left (-\frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \relax (2)^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \relax (2)^{2}}{\log \relax (2)^{4}}\right )} \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 -\frac {2 \, {\left (15 \, x^{9} - x e^{\left (x^{2} + \frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \relax (2)^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \relax (2)^{2}}{\log \relax (2)^{4}}\right )} \log \relax (2)^{4} + 6 \, {\left (x^{3} - 6 \, x^{2} + 8 \, x\right )} \log \relax (2)^{4} + 3 \, {\left (7 \, x^{6} - 24 \, x^{5}\right )} \log \relax (2)^{2}\right )} e^{\left (-\frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \relax (2)^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \relax (2)^{2}}{\log \relax (2)^{4}}\right )}}{\log \relax (2)^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 36, normalized size = 1.16
method | result | size |
risch | \({\mathrm e}^{x^{2}}+3 \,{\mathrm e}^{-\frac {x^{2} \left (x^{4}+x \ln \relax (2)^{2}-4 \ln \relax (2)^{2}\right )^{2}}{\ln \relax (2)^{4}}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 65, normalized size = 2.10 \begin {gather*} \frac {3 \, e^{\left (-\frac {x^{10}}{\log \relax (2)^{4}} - \frac {2 \, x^{7}}{\log \relax (2)^{2}} - x^{4} + \frac {8 \, x^{6}}{\log \relax (2)^{2}} + 8 \, x^{3} - 16 \, x^{2}\right )} \log \relax (2)^{4} + e^{\left (x^{2}\right )} \log \relax (2)^{4}}{\log \relax (2)^{4}} \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 {{\mathrm {e}}^{-\frac {{\ln \relax (2)}^4\,\left (x^4-8\,x^3+16\,x^2\right )+x^{10}-{\ln \relax (2)}^2\,\left (8\,x^6-2\,x^7\right )}{{\ln \relax (2)}^4}}\,\left ({\ln \relax (2)}^4\,\left (12\,x^3-72\,x^2+96\,x\right )+30\,x^9-{\ln \relax (2)}^2\,\left (144\,x^5-42\,x^6\right )-2\,x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {{\ln \relax (2)}^4\,\left (x^4-8\,x^3+16\,x^2\right )+x^{10}-{\ln \relax (2)}^2\,\left (8\,x^6-2\,x^7\right )}{{\ln \relax (2)}^4}}\,{\ln \relax (2)}^4\right )}{{\ln \relax (2)}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 49, normalized size = 1.58 \begin {gather*} e^{x^{2}} + 3 e^{- \frac {x^{10} + \left (2 x^{7} - 8 x^{6}\right ) \log {\relax (2 )}^{2} + \left (x^{4} - 8 x^{3} + 16 x^{2}\right ) \log {\relax (2 )}^{4}}{\log {\relax (2 )}^{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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