Optimal. Leaf size=29 \[ \frac {e^{2 x}}{\left (5+e^{2 x^2}\right ) \left (-x+4 \log ^2(5)\right )} \]
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Rubi [F] time = 2.63, 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 {e^{2 x} \left (5-10 x+40 \log ^2(5)\right )+e^{2 x+2 x^2} \left (1-2 x+4 x^2+(8-16 x) \log ^2(5)\right )}{25 x^2-200 x \log ^2(5)+400 \log ^4(5)+e^{4 x^2} \left (x^2-8 x \log ^2(5)+16 \log ^4(5)\right )+e^{2 x^2} \left (10 x^2-80 x \log ^2(5)+160 \log ^4(5)\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 {e^{2 x} \left (-10 x+5 \left (1+8 \log ^2(5)\right )+e^{2 x^2} \left (1+4 x^2+8 \log ^2(5)-2 x \left (1+8 \log ^2(5)\right )\right )\right )}{\left (5+e^{2 x^2}\right )^2 \left (x-4 \log ^2(5)\right )^2} \, dx\\ &=\int \left (-\frac {20 e^{2 x} x}{\left (5+e^{2 x^2}\right )^2 \left (x-4 \log ^2(5)\right )}+\frac {e^{2 x} \left (1+4 x^2+8 \log ^2(5)-2 x \left (1+8 \log ^2(5)\right )\right )}{\left (5+e^{2 x^2}\right ) \left (x-4 \log ^2(5)\right )^2}\right ) \, dx\\ &=-\left (20 \int \frac {e^{2 x} x}{\left (5+e^{2 x^2}\right )^2 \left (x-4 \log ^2(5)\right )} \, dx\right )+\int \frac {e^{2 x} \left (1+4 x^2+8 \log ^2(5)-2 x \left (1+8 \log ^2(5)\right )\right )}{\left (5+e^{2 x^2}\right ) \left (x-4 \log ^2(5)\right )^2} \, dx\\ &=-\left (20 \int \left (\frac {e^{2 x}}{\left (5+e^{2 x^2}\right )^2}+\frac {4 e^{2 x} \log ^2(5)}{\left (5+e^{2 x^2}\right )^2 \left (x-4 \log ^2(5)\right )}\right ) \, dx\right )+\int \left (\frac {4 e^{2 x}}{5+e^{2 x^2}}+\frac {e^{2 x}}{\left (5+e^{2 x^2}\right ) \left (x-4 \log ^2(5)\right )^2}+\frac {2 e^{2 x} \left (-1+8 \log ^2(5)\right )}{\left (5+e^{2 x^2}\right ) \left (x-4 \log ^2(5)\right )}\right ) \, dx\\ &=4 \int \frac {e^{2 x}}{5+e^{2 x^2}} \, dx-20 \int \frac {e^{2 x}}{\left (5+e^{2 x^2}\right )^2} \, dx-\left (80 \log ^2(5)\right ) \int \frac {e^{2 x}}{\left (5+e^{2 x^2}\right )^2 \left (x-4 \log ^2(5)\right )} \, dx-\left (2 \left (1-8 \log ^2(5)\right )\right ) \int \frac {e^{2 x}}{\left (5+e^{2 x^2}\right ) \left (x-4 \log ^2(5)\right )} \, dx+\int \frac {e^{2 x}}{\left (5+e^{2 x^2}\right ) \left (x-4 \log ^2(5)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.25, size = 28, normalized size = 0.97 \begin {gather*} -\frac {e^{2 x}}{\left (5+e^{2 x^2}\right ) \left (x-4 \log ^2(5)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 49, normalized size = 1.69 \begin {gather*} \frac {e^{\left (2 \, x^{2} + 2 \, x\right )}}{{\left (4 \, \log \relax (5)^{2} - x\right )} e^{\left (4 \, x^{2}\right )} + 5 \, {\left (4 \, \log \relax (5)^{2} - x\right )} e^{\left (2 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 38, normalized size = 1.31 \begin {gather*} \frac {e^{\left (2 \, x\right )}}{4 \, e^{\left (2 \, x^{2}\right )} \log \relax (5)^{2} - x e^{\left (2 \, x^{2}\right )} + 20 \, \log \relax (5)^{2} - 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 28, normalized size = 0.97
method | result | size |
norman | \(\frac {{\mathrm e}^{2 x}}{\left (4 \ln \relax (5)^{2}-x \right ) \left (5+{\mathrm e}^{2 x^{2}}\right )}\) | \(28\) |
risch | \(\frac {{\mathrm e}^{2 x}}{\left (4 \ln \relax (5)^{2}-x \right ) \left (5+{\mathrm e}^{2 x^{2}}\right )}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 34, normalized size = 1.17 \begin {gather*} \frac {e^{\left (2 \, x\right )}}{{\left (4 \, \log \relax (5)^{2} - x\right )} e^{\left (2 \, x^{2}\right )} + 20 \, \log \relax (5)^{2} - 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.96, size = 45, normalized size = 1.55 \begin {gather*} -\frac {x^2\,{\mathrm {e}}^{2\,x}-4\,x\,{\mathrm {e}}^{2\,x}\,{\ln \relax (5)}^2}{x\,{\left (x-4\,{\ln \relax (5)}^2\right )}^2\,\left ({\mathrm {e}}^{2\,x^2}+5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 31, normalized size = 1.07 \begin {gather*} - \frac {e^{2 x}}{5 x + \left (x - 4 \log {\relax (5 )}^{2}\right ) e^{2 x^{2}} - 20 \log {\relax (5 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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