Optimal. Leaf size=23 \[ e^{-x^2} \left (\frac {841}{25}+x-4 (x+\log (5))^2\right )^2 \]
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Rubi [C] time = 0.27, antiderivative size = 241, normalized size of antiderivative = 10.48, number of steps used = 16, number of rules used = 6, integrand size = 98, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6, 12, 2226, 2212, 2209, 2205} \begin {gather*} -\frac {1}{25} \sqrt {\pi } (1-8 \log (5)) \left (991-100 \log ^2(5)\right ) \text {erf}(x)+\frac {1}{25} \sqrt {\pi } \left (841+800 \log ^3(5)-100 \log ^2(5)-6728 \log (5)\right ) \text {erf}(x)+6 \sqrt {\pi } (1-8 \log (5)) \text {erf}(x)+32 e^{-x^2} x^2+32 e^{-x^2}-\frac {1}{25} e^{-x^2} x^2 \left (7503-2400 \log ^2(5)+400 \log (5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )-\frac {1}{25} e^{-x^2} \left (7503-2400 \log ^2(5)+400 \log (5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log ^4(5)-28525 \log ^2(5)+1250 \log (5)\right )-12 e^{-x^2} x (1-8 \log (5))+16 e^{-x^2} x^4-8 e^{-x^2} x^3 (1-8 \log (5)) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2205
Rule 2209
Rule 2212
Rule 2226
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{625} e^{-x^2} \left (42050-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)+x \left (-1749712-20000 \log ^4(5)\right )\right ) \, dx\\ &=\frac {1}{625} \int e^{-x^2} \left (42050-99100 x^2+375150 x^3+10000 x^4-20000 x^5+\left (-336400-20000 x+792800 x^2+20000 x^3-80000 x^4\right ) \log (5)+\left (-5000+456400 x+10000 x^2-120000 x^3\right ) \log ^2(5)+\left (40000-80000 x^2\right ) \log ^3(5)+x \left (-1749712-20000 \log ^4(5)\right )\right ) \, dx\\ &=\frac {1}{625} \int \left (-20000 e^{-x^2} x^5-10000 e^{-x^2} x^4 (-1+8 \log (5))+100 e^{-x^2} x^2 (1-8 \log (5)) \left (-991+100 \log ^2(5)\right )-50 e^{-x^2} x^3 \left (-7503-400 \log (5)+2400 \log ^2(5)\right )+50 e^{-x^2} \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )-16 e^{-x^2} x \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )\right ) \, dx\\ &=-\left (32 \int e^{-x^2} x^5 \, dx\right )+(16 (1-8 \log (5))) \int e^{-x^2} x^4 \, dx+\frac {1}{25} \left (2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )\right ) \int e^{-x^2} x^3 \, dx-\frac {1}{25} \left (4 (1-8 \log (5)) \left (991-100 \log ^2(5)\right )\right ) \int e^{-x^2} x^2 \, dx+\frac {1}{25} \left (2 \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )\right ) \int e^{-x^2} \, dx-\frac {1}{625} \left (16 \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )\right ) \int e^{-x^2} x \, dx\\ &=16 e^{-x^2} x^4-8 e^{-x^2} x^3 (1-8 \log (5))-\frac {1}{25} e^{-x^2} x^2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )+\frac {1}{25} \sqrt {\pi } \text {erf}(x) \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )-64 \int e^{-x^2} x^3 \, dx+(24 (1-8 \log (5))) \int e^{-x^2} x^2 \, dx+\frac {1}{25} \left (2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )\right ) \int e^{-x^2} x \, dx-\frac {1}{25} \left (2 (1-8 \log (5)) \left (991-100 \log ^2(5)\right )\right ) \int e^{-x^2} \, dx\\ &=32 e^{-x^2} x^2+16 e^{-x^2} x^4-12 e^{-x^2} x (1-8 \log (5))-8 e^{-x^2} x^3 (1-8 \log (5))-\frac {1}{25} e^{-x^2} \left (7503+400 \log (5)-2400 \log ^2(5)\right )-\frac {1}{25} e^{-x^2} x^2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )-\frac {1}{25} \sqrt {\pi } \text {erf}(x) (1-8 \log (5)) \left (991-100 \log ^2(5)\right )+\frac {1}{25} \sqrt {\pi } \text {erf}(x) \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )-64 \int e^{-x^2} x \, dx+(12 (1-8 \log (5))) \int e^{-x^2} \, dx\\ &=32 e^{-x^2}+32 e^{-x^2} x^2+16 e^{-x^2} x^4-12 e^{-x^2} x (1-8 \log (5))-8 e^{-x^2} x^3 (1-8 \log (5))+6 \sqrt {\pi } \text {erf}(x) (1-8 \log (5))-\frac {1}{25} e^{-x^2} \left (7503+400 \log (5)-2400 \log ^2(5)\right )-\frac {1}{25} e^{-x^2} x^2 \left (7503+400 \log (5)-2400 \log ^2(5)\right )+\frac {2}{25} e^{-x^2} x (1-8 \log (5)) \left (991-100 \log ^2(5)\right )-\frac {1}{25} \sqrt {\pi } \text {erf}(x) (1-8 \log (5)) \left (991-100 \log ^2(5)\right )+\frac {1}{25} \sqrt {\pi } \text {erf}(x) \left (841-6728 \log (5)-100 \log ^2(5)+800 \log ^3(5)\right )+\frac {8}{625} e^{-x^2} \left (109357+1250 \log (5)-28525 \log ^2(5)+1250 \log ^4(5)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 35, normalized size = 1.52 \begin {gather*} \frac {1}{625} e^{-x^2} \left (-841+100 x^2+100 \log ^2(5)+25 x (-1+8 \log (5))\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 75, normalized size = 3.26 \begin {gather*} \frac {1}{625} \, {\left (10000 \, x^{4} + 40000 \, x \log \relax (5)^{3} + 10000 \, \log \relax (5)^{4} - 5000 \, x^{3} + 200 \, {\left (300 \, x^{2} - 25 \, x - 841\right )} \log \relax (5)^{2} - 167575 \, x^{2} + 400 \, {\left (100 \, x^{3} - 25 \, x^{2} - 841 \, x\right )} \log \relax (5) + 42050 \, x + 707281\right )} e^{\left (-x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 82, normalized size = 3.57 \begin {gather*} \frac {1}{625} \, {\left (10000 \, x^{4} + 40000 \, x^{3} \log \relax (5) + 60000 \, x^{2} \log \relax (5)^{2} + 40000 \, x \log \relax (5)^{3} + 10000 \, \log \relax (5)^{4} - 5000 \, x^{3} - 10000 \, x^{2} \log \relax (5) - 5000 \, x \log \relax (5)^{2} - 167575 \, x^{2} - 336400 \, x \log \relax (5) - 168200 \, \log \relax (5)^{2} + 42050 \, x + 707281\right )} e^{\left (-x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 32, normalized size = 1.39
method | result | size |
gosper | \(\frac {\left (100 \ln \relax (5)^{2}+200 x \ln \relax (5)+100 x^{2}-25 x -841\right )^{2} {\mathrm e}^{-x^{2}}}{625}\) | \(32\) |
norman | \(\left (\left (64 \ln \relax (5)-8\right ) x^{3}+\left (-\frac {6703}{25}+96 \ln \relax (5)^{2}-16 \ln \relax (5)\right ) x^{2}+\left (64 \ln \relax (5)^{3}-8 \ln \relax (5)^{2}-\frac {13456 \ln \relax (5)}{25}+\frac {1682}{25}\right ) x +16 x^{4}+\frac {707281}{625}-\frac {6728 \ln \relax (5)^{2}}{25}+16 \ln \relax (5)^{4}\right ) {\mathrm e}^{-x^{2}}\) | \(73\) |
risch | \(\frac {\left (10000 \ln \relax (5)^{4}+40000 \ln \relax (5)^{3} x +60000 x^{2} \ln \relax (5)^{2}+40000 x^{3} \ln \relax (5)+10000 x^{4}-5000 x \ln \relax (5)^{2}-10000 x^{2} \ln \relax (5)-5000 x^{3}-168200 \ln \relax (5)^{2}-336400 x \ln \relax (5)-167575 x^{2}+42050 x +707281\right ) {\mathrm e}^{-x^{2}}}{625}\) | \(83\) |
default | \(\frac {707281 \,{\mathrm e}^{-x^{2}}}{625}+\frac {1682 x \,{\mathrm e}^{-x^{2}}}{25}-\frac {6703 \,{\mathrm e}^{-x^{2}} x^{2}}{25}-8 \,{\mathrm e}^{-x^{2}} x^{3}+16 \,{\mathrm e}^{-x^{2}} x^{4}-\frac {6728 \,{\mathrm e}^{-x^{2}} \ln \relax (5)^{2}}{25}+16 \ln \relax (5)^{4} {\mathrm e}^{-x^{2}}-\frac {13456 \,{\mathrm e}^{-x^{2}} x \ln \relax (5)}{25}-8 \,{\mathrm e}^{-x^{2}} x \ln \relax (5)^{2}+64 \ln \relax (5)^{3} x \,{\mathrm e}^{-x^{2}}-16 \,{\mathrm e}^{-x^{2}} x^{2} \ln \relax (5)+96 \,{\mathrm e}^{-x^{2}} x^{2} \ln \relax (5)^{2}+64 \,{\mathrm e}^{-x^{2}} x^{3} \ln \relax (5)\) | \(154\) |
meijerg | \(\frac {841 \sqrt {\pi }\, \erf \relax (x )}{25}+\frac {\left (-128 \ln \relax (5)+16\right ) \left (-\frac {x \left (10 x^{2}+15\right ) {\mathrm e}^{-x^{2}}}{10}+\frac {3 \sqrt {\pi }\, \erf \relax (x )}{4}\right )}{2}+\frac {\left (-128 \ln \relax (5)^{3}+16 \ln \relax (5)^{2}+\frac {31712 \ln \relax (5)}{25}-\frac {3964}{25}\right ) \left (-x \,{\mathrm e}^{-x^{2}}+\frac {\sqrt {\pi }\, \erf \relax (x )}{2}\right )}{2}-32+\frac {16 \left (3 x^{4}+6 x^{2}+6\right ) {\mathrm e}^{-x^{2}}}{3}+\frac {\left (-192 \ln \relax (5)^{2}+32 \ln \relax (5)+\frac {15006}{25}\right ) \left (1-\frac {\left (2 x^{2}+2\right ) {\mathrm e}^{-x^{2}}}{2}\right )}{2}+\frac {\left (-32 \ln \relax (5)^{4}+\frac {18256 \ln \relax (5)^{2}}{25}-32 \ln \relax (5)-\frac {1749712}{625}\right ) \left (1-{\mathrm e}^{-x^{2}}\right )}{2}+32 \ln \relax (5)^{3} \sqrt {\pi }\, \erf \relax (x )-4 \ln \relax (5)^{2} \sqrt {\pi }\, \erf \relax (x )-\frac {6728 \ln \relax (5) \sqrt {\pi }\, \erf \relax (x )}{25}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.47, size = 259, normalized size = 11.26 \begin {gather*} 16 \, e^{\left (-x^{2}\right )} \log \relax (5)^{4} + 32 \, \sqrt {\pi } \operatorname {erf}\relax (x) \log \relax (5)^{3} + 96 \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \log \relax (5)^{2} + 32 \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\relax (x)\right )} \log \relax (5)^{3} - 4 \, \sqrt {\pi } \operatorname {erf}\relax (x) \log \relax (5)^{2} - 16 \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \log \relax (5) - 4 \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\relax (x)\right )} \log \relax (5)^{2} - \frac {9128}{25} \, e^{\left (-x^{2}\right )} \log \relax (5)^{2} - \frac {6728}{25} \, \sqrt {\pi } \operatorname {erf}\relax (x) \log \relax (5) + 16 \, {\left (x^{4} + 2 \, x^{2} + 2\right )} e^{\left (-x^{2}\right )} - 4 \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} - \frac {7503}{25} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} + \frac {1982}{25} \, x e^{\left (-x^{2}\right )} + 16 \, {\left (2 \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} - 3 \, \sqrt {\pi } \operatorname {erf}\relax (x)\right )} \log \relax (5) - \frac {7928}{25} \, {\left (2 \, x e^{\left (-x^{2}\right )} - \sqrt {\pi } \operatorname {erf}\relax (x)\right )} \log \relax (5) + 16 \, e^{\left (-x^{2}\right )} \log \relax (5) + \frac {874856}{625} \, e^{\left (-x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 31, normalized size = 1.35 \begin {gather*} \frac {{\mathrm {e}}^{-x^2}\,{\left (200\,x\,\ln \relax (5)-25\,x+100\,{\ln \relax (5)}^2+100\,x^2-841\right )}^2}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.26, size = 90, normalized size = 3.91 \begin {gather*} \frac {\left (10000 x^{4} - 5000 x^{3} + 40000 x^{3} \log {\relax (5 )} - 167575 x^{2} - 10000 x^{2} \log {\relax (5 )} + 60000 x^{2} \log {\relax (5 )}^{2} - 336400 x \log {\relax (5 )} - 5000 x \log {\relax (5 )}^{2} + 42050 x + 40000 x \log {\relax (5 )}^{3} - 168200 \log {\relax (5 )}^{2} + 10000 \log {\relax (5 )}^{4} + 707281\right ) e^{- x^{2}}}{625} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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