Optimal. Leaf size=34 \[ 5-x-\frac {\left (2+\frac {1}{2+4 x+e^{x^2} x-\log ^2(5)}\right )^2}{x^2} \]
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Rubi [F] time = 6.89, 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 {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{8 x^3+48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)-x^3 \log ^6(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100+560 x+960 x^2+504 x^3-48 x^4-96 x^5-64 x^6+e^{3 x^2} \left (8 x^3-x^6\right )+\left (-130-464 x-384 x^2+12 x^3+48 x^4+48 x^5\right ) \log ^2(5)+\left (56+96 x-6 x^3-12 x^4\right ) \log ^4(5)+\left (-8+x^3\right ) \log ^6(5)+e^{2 x^2} \left (60 x^2+96 x^3+8 x^4-6 x^5-12 x^6+\left (-24 x^2+3 x^5\right ) \log ^2(5)\right )+e^{x^2} \left (140 x+480 x^2+404 x^3+20 x^4-48 x^5-48 x^6+\left (-116 x-192 x^2-8 x^3+12 x^4+24 x^5\right ) \log ^2(5)+\left (24 x-3 x^4\right ) \log ^4(5)\right )}{48 x^4+96 x^5+64 x^6+e^{3 x^2} x^6+\left (-12 x^3-48 x^4-48 x^5\right ) \log ^2(5)+\left (6 x^3+12 x^4\right ) \log ^4(5)+e^{2 x^2} \left (6 x^5+12 x^6-3 x^5 \log ^2(5)\right )+e^{x^2} \left (12 x^4+48 x^5+48 x^6+\left (-12 x^4-24 x^5\right ) \log ^2(5)+3 x^4 \log ^4(5)\right )+x^3 \left (8-\log ^6(5)\right )} \, dx\\ &=\int \frac {-\left (4+e^{x^2}\right )^3 x^6+3 \left (4+e^{x^2}\right )^2 x^5 \left (-2+\log ^2(5)\right )-2 \left (5-2 \log ^2(5)\right )^2 \left (-2+\log ^2(5)\right )-12 \left (4+e^{x^2}\right )^2 x^2 \left (-5+2 \log ^2(5)\right )+4 \left (4+e^{x^2}\right ) x \left (35-29 \log ^2(5)+6 \log ^4(5)\right )+x^3 \left (504+96 e^{2 x^2}+8 e^{3 x^2}+12 \log ^2(5)-6 \log ^4(5)+\log ^6(5)+e^{x^2} \left (404-8 \log ^2(5)\right )\right )+x^4 \left (8 e^{2 x^2}-12 \left (-2+\log ^2(5)\right )^2+e^{x^2} \left (20+12 \log ^2(5)-3 \log ^4(5)\right )\right )}{x^3 \left (\left (4+e^{x^2}\right ) x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3} \, dx\\ &=\int \left (\frac {8-x^3}{x^3}+\frac {2 \left (-2-8 x^3+\log ^2(5)-2 x^2 \left (2-\log ^2(5)\right )\right )}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}+\frac {4 \left (-1-8 x^3+\log ^2(5)-x^2 \left (3-2 \log ^2(5)\right )\right )}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}+\frac {4 \left (3+2 x^2\right )}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )}\right ) \, dx\\ &=2 \int \frac {-2-8 x^3+\log ^2(5)-2 x^2 \left (2-\log ^2(5)\right )}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3} \, dx+4 \int \frac {-1-8 x^3+\log ^2(5)-x^2 \left (3-2 \log ^2(5)\right )}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2} \, dx+4 \int \frac {3+2 x^2}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )} \, dx+\int \frac {8-x^3}{x^3} \, dx\\ &=2 \int \left (\frac {8}{\left (-4 x-e^{x^2} x-2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}+\frac {-2+\log ^2(5)}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}+\frac {2 \left (-2+\log ^2(5)\right )}{x \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3}\right ) \, dx+4 \int \left (-\frac {8}{\left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}+\frac {-1+\log ^2(5)}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}+\frac {-3+2 \log ^2(5)}{x \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2}\right ) \, dx+4 \int \left (\frac {3}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )}+\frac {2}{x \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )}\right ) \, dx+\int \left (-1+\frac {8}{x^3}\right ) \, dx\\ &=-\frac {4}{x^2}-x+8 \int \frac {1}{x \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )} \, dx+12 \int \frac {1}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )} \, dx+16 \int \frac {1}{\left (-4 x-e^{x^2} x-2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3} \, dx-32 \int \frac {1}{\left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2} \, dx-\left (4 \left (3-2 \log ^2(5)\right )\right ) \int \frac {1}{x \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2} \, dx-\left (4 \left (1-\log ^2(5)\right )\right ) \int \frac {1}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^2} \, dx-\left (2 \left (2-\log ^2(5)\right )\right ) \int \frac {1}{x^3 \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3} \, dx-\left (4 \left (2-\log ^2(5)\right )\right ) \int \frac {1}{x \left (4 x+e^{x^2} x+2 \left (1-\frac {\log ^2(5)}{2}\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.26, size = 49, normalized size = 1.44 \begin {gather*} -\frac {x^3+\frac {\left (5+2 \left (4+e^{x^2}\right ) x-2 \log ^2(5)\right )^2}{\left (2+\left (4+e^{x^2}\right ) x-\log ^2(5)\right )^2}}{x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 186, normalized size = 5.47 \begin {gather*} -\frac {16 \, x^{5} + {\left (x^{3} + 4\right )} \log \relax (5)^{4} + 16 \, x^{4} + 4 \, x^{3} - 4 \, {\left (2 \, x^{4} + x^{3} + 8 \, x + 5\right )} \log \relax (5)^{2} + 64 \, x^{2} + {\left (x^{5} + 4 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (4 \, x^{5} + 2 \, x^{4} - {\left (x^{4} + 4 \, x\right )} \log \relax (5)^{2} + 16 \, x^{2} + 10 \, x\right )} e^{\left (x^{2}\right )} + 80 \, x + 25}{x^{2} \log \relax (5)^{4} + x^{4} e^{\left (2 \, x^{2}\right )} + 16 \, x^{4} + 16 \, x^{3} - 4 \, {\left (2 \, x^{3} + x^{2}\right )} \log \relax (5)^{2} + 4 \, x^{2} - 2 \, {\left (x^{3} \log \relax (5)^{2} - 4 \, x^{4} - 2 \, x^{3}\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 = 0.72, size = 240, normalized size = 7.06 \begin {gather*} -\frac {2 \, x^{4} e^{\left (x^{2}\right )} \log \relax (5)^{2} - x^{3} \log \relax (5)^{4} - x^{5} e^{\left (2 \, x^{2}\right )} - 8 \, x^{5} e^{\left (x^{2}\right )} + 8 \, x^{4} \log \relax (5)^{2} - 16 \, x^{5} - 4 \, x^{4} e^{\left (x^{2}\right )} + 4 \, x^{3} \log \relax (5)^{2} - 16 \, x^{4} + 8 \, x e^{\left (x^{2}\right )} \log \relax (5)^{2} - 4 \, \log \relax (5)^{4} - 4 \, x^{3} - 4 \, x^{2} e^{\left (2 \, x^{2}\right )} - 32 \, x^{2} e^{\left (x^{2}\right )} + 32 \, x \log \relax (5)^{2} - 64 \, x^{2} - 20 \, x e^{\left (x^{2}\right )} + 20 \, \log \relax (5)^{2} - 80 \, x - 25}{2 \, x^{3} e^{\left (x^{2}\right )} \log \relax (5)^{2} - x^{2} \log \relax (5)^{4} - x^{4} e^{\left (2 \, x^{2}\right )} - 8 \, x^{4} e^{\left (x^{2}\right )} + 8 \, x^{3} \log \relax (5)^{2} - 16 \, x^{4} - 4 \, x^{3} e^{\left (x^{2}\right )} + 4 \, x^{2} \log \relax (5)^{2} - 16 \, x^{3} - 4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 50, normalized size = 1.47
method | result | size |
risch | \(-x -\frac {4}{x^{2}}+\frac {4 \ln \relax (5)^{2}-4 \,{\mathrm e}^{x^{2}} x -16 x -9}{x^{2} \left (\ln \relax (5)^{2}-{\mathrm e}^{x^{2}} x -4 x -2\right )^{2}}\) | \(50\) |
norman | \(\frac {\left (-80+32 \ln \relax (5)^{2}\right ) x -64 x^{2}+\left (8 \ln \relax (5)^{2}-16\right ) x^{4}+\left (-\ln \relax (5)^{4}+4 \ln \relax (5)^{2}-4\right ) x^{3}+\left (-20+8 \ln \relax (5)^{2}\right ) x \,{\mathrm e}^{x^{2}}+\left (2 \ln \relax (5)^{2}-4\right ) x^{4} {\mathrm e}^{x^{2}}-25-16 x^{5}-32 x^{2} {\mathrm e}^{x^{2}}-4 x^{2} {\mathrm e}^{2 x^{2}}-8 \,{\mathrm e}^{x^{2}} x^{5}-{\mathrm e}^{2 x^{2}} x^{5}-4 \ln \relax (5)^{4}+20 \ln \relax (5)^{2}}{x^{2} \left (\ln \relax (5)^{2}-{\mathrm e}^{x^{2}} x -4 x -2\right )^{2}}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 186, normalized size = 5.47 \begin {gather*} \frac {8 \, {\left (\log \relax (5)^{2} - 2\right )} x^{4} - 16 \, x^{5} - {\left (\log \relax (5)^{4} - 4 \, \log \relax (5)^{2} + 4\right )} x^{3} - 4 \, \log \relax (5)^{4} + 16 \, {\left (2 \, \log \relax (5)^{2} - 5\right )} x - 64 \, x^{2} - {\left (x^{5} + 4 \, x^{2}\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left ({\left (\log \relax (5)^{2} - 2\right )} x^{4} - 4 \, x^{5} + 2 \, {\left (2 \, \log \relax (5)^{2} - 5\right )} x - 16 \, x^{2}\right )} e^{\left (x^{2}\right )} + 20 \, \log \relax (5)^{2} - 25}{x^{4} e^{\left (2 \, x^{2}\right )} - 8 \, {\left (\log \relax (5)^{2} - 2\right )} x^{3} + 16 \, x^{4} + {\left (\log \relax (5)^{4} - 4 \, \log \relax (5)^{2} + 4\right )} x^{2} - 2 \, {\left ({\left (\log \relax (5)^{2} - 2\right )} x^{3} - 4 \, x^{4}\right )} e^{\left (x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 213, normalized size = 6.26 \begin {gather*} -x-\frac {4}{x^2}-\frac {4\,\left (4\,x^2-{\ln \relax (5)}^2-2\,x^2\,{\ln \relax (5)}^2+8\,x^3+2\right )}{x\,\left ({\mathrm {e}}^{x^2}+\frac {4\,x-{\ln \relax (5)}^2+2}{x}\right )\,\left (2\,x^2-2\,x^4\,{\ln \relax (5)}^2-x^2\,{\ln \relax (5)}^2+4\,x^4+8\,x^5\right )}-\frac {4\,x^2-{\ln \relax (5)}^2-2\,x^2\,{\ln \relax (5)}^2+8\,x^3+2}{x^2\,\left ({\mathrm {e}}^{2\,x^2}+\frac {{\left (4\,x-{\ln \relax (5)}^2+2\right )}^2}{x^2}+\frac {2\,{\mathrm {e}}^{x^2}\,\left (4\,x-{\ln \relax (5)}^2+2\right )}{x}\right )\,\left (2\,x^2-2\,x^4\,{\ln \relax (5)}^2-x^2\,{\ln \relax (5)}^2+4\,x^4+8\,x^5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.53, size = 105, normalized size = 3.09 \begin {gather*} - x + \frac {- 4 x e^{x^{2}} - 16 x - 9 + 4 \log {\relax (5 )}^{2}}{x^{4} e^{2 x^{2}} + 16 x^{4} - 8 x^{3} \log {\relax (5 )}^{2} + 16 x^{3} - 4 x^{2} \log {\relax (5 )}^{2} + 4 x^{2} + x^{2} \log {\relax (5 )}^{4} + \left (8 x^{4} - 2 x^{3} \log {\relax (5 )}^{2} + 4 x^{3}\right ) e^{x^{2}}} - \frac {4}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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