Optimal. Leaf size=29 \[ 4+\log \left (2+\frac {3}{-3+e^2-5 e^{-x}-x^2+\log (\log (4))}\right ) \]
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Rubi [A] time = 3.35, antiderivative size = 55, normalized size of antiderivative = 1.90, number of steps used = 6, number of rules used = 4, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6741, 12, 6742, 6684} \begin {gather*} \log \left (2 e^x x^2+e^x \left (3-2 \left (e^2+\log (\log (4))\right )\right )+10\right )-\log \left (e^x x^2+e^x \left (3-e^2-\log (\log (4))\right )+5\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6684
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^x \left (-5+2 e^x x\right )}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx\\ &=3 \int \frac {e^x \left (-5+2 e^x x\right )}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx\\ &=3 \int \left (\frac {e^x \left (-3+e^2-2 x-x^2+\log (\log (4))\right )}{3 \left (5+e^x x^2+3 e^x \left (1+\frac {1}{3} \left (-e^2-\log (\log (4))\right )\right )\right )}+\frac {e^x \left (3-2 e^2+4 x+2 x^2-2 \log (\log (4))\right )}{3 \left (10+2 e^x x^2+3 e^x \left (1-\frac {2}{3} \left (e^2+\log (\log (4))\right )\right )\right )}\right ) \, dx\\ &=\int \frac {e^x \left (-3+e^2-2 x-x^2+\log (\log (4))\right )}{5+e^x x^2+3 e^x \left (1+\frac {1}{3} \left (-e^2-\log (\log (4))\right )\right )} \, dx+\int \frac {e^x \left (3-2 e^2+4 x+2 x^2-2 \log (\log (4))\right )}{10+2 e^x x^2+3 e^x \left (1-\frac {2}{3} \left (e^2+\log (\log (4))\right )\right )} \, dx\\ &=-\log \left (5+e^x x^2+e^x \left (3-e^2-\log (\log (4))\right )\right )+\log \left (10+2 e^x x^2+e^x \left (3-2 \left (e^2+\log (\log (4))\right )\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 3.63, size = 72, normalized size = 2.48 \begin {gather*} 3 \left (\frac {1}{3} \log \left (10+3 e^x-2 e^{2+x}+2 e^x x^2-2 e^x \log (\log (4))\right )-\frac {1}{3} \log \left (5+3 e^x-e^{2+x}+e^x x^2-e^x \log (\log (4))\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 133, normalized size = 4.59 \begin {gather*} \log \left (2 \, x^{2} - 2 \, e^{2} - 2 \, \log \left (2 \, \log \relax (2)\right ) + 3\right ) - \log \left (x^{2} - e^{2} - \log \left (2 \, \log \relax (2)\right ) + 3\right ) + \log \left (-\frac {{\left (2 \, x^{2} - 2 \, e^{2} + 3\right )} e^{x} - 2 \, e^{x} \log \left (2 \, \log \relax (2)\right ) + 10}{2 \, x^{2} - 2 \, e^{2} - 2 \, \log \left (2 \, \log \relax (2)\right ) + 3}\right ) - \log \left (-\frac {{\left (x^{2} - e^{2} + 3\right )} e^{x} - e^{x} \log \left (2 \, \log \relax (2)\right ) + 5}{x^{2} - e^{2} - \log \left (2 \, \log \relax (2)\right ) + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 68, normalized size = 2.34 \begin {gather*} \log \left (2 \, x^{2} e^{x} - 2 \, e^{x} \log \relax (2) - 2 \, e^{x} \log \left (\log \relax (2)\right ) - 2 \, e^{\left (x + 2\right )} + 3 \, e^{x} + 10\right ) - \log \left (x^{2} e^{x} - e^{x} \log \relax (2) - e^{x} \log \left (\log \relax (2)\right ) - e^{\left (x + 2\right )} + 3 \, e^{x} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.77, size = 60, normalized size = 2.07
method | result | size |
norman | \(-\ln \left (-{\mathrm e}^{x} x^{2}+{\mathrm e}^{2} {\mathrm e}^{x}+{\mathrm e}^{x} \ln \left (2 \ln \relax (2)\right )-3 \,{\mathrm e}^{x}-5\right )+\ln \left (-2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{2} {\mathrm e}^{x}+2 \,{\mathrm e}^{x} \ln \left (2 \ln \relax (2)\right )-3 \,{\mathrm e}^{x}-10\right )\) | \(60\) |
risch | \(-\ln \left (x^{2}-{\mathrm e}^{2}-\ln \left (\ln \relax (2)\right )-\ln \relax (2)+3\right )+\ln \left (-2 x^{2}+2 \,{\mathrm e}^{2}+2 \ln \left (\ln \relax (2)\right )+2 \ln \relax (2)-3\right )+\ln \left ({\mathrm e}^{x}-\frac {10}{-2 x^{2}+2 \,{\mathrm e}^{2}+2 \ln \left (\ln \relax (2)\right )+2 \ln \relax (2)-3}\right )-\ln \left ({\mathrm e}^{x}-\frac {5}{-x^{2}+{\mathrm e}^{2}+\ln \left (\ln \relax (2)\right )+\ln \relax (2)-3}\right )\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 139, normalized size = 4.79 \begin {gather*} \log \left (2 \, x^{2} - 2 \, e^{2} - 2 \, \log \relax (2) - 2 \, \log \left (\log \relax (2)\right ) + 3\right ) - \log \left (x^{2} - e^{2} - \log \relax (2) - \log \left (\log \relax (2)\right ) + 3\right ) + \log \left (\frac {{\left (2 \, x^{2} - 2 \, e^{2} - 2 \, \log \relax (2) - 2 \, \log \left (\log \relax (2)\right ) + 3\right )} e^{x} + 10}{2 \, x^{2} - 2 \, e^{2} - 2 \, \log \relax (2) - 2 \, \log \left (\log \relax (2)\right ) + 3}\right ) - \log \left (\frac {{\left (x^{2} - e^{2} - \log \relax (2) - \log \left (\log \relax (2)\right ) + 3\right )} e^{x} + 5}{x^{2} - e^{2} - \log \relax (2) - \log \left (\log \relax (2)\right ) + 3}\right ) \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 {15\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^{2\,x}}{2\,{\ln \left (2\,\ln \relax (2)\right )}^2\,{\mathrm {e}}^{2\,x}-\ln \left (2\,\ln \relax (2)\right )\,\left (20\,{\mathrm {e}}^x+{\mathrm {e}}^{2\,x}\,\left (4\,x^2-4\,{\mathrm {e}}^2+9\right )\right )+{\mathrm {e}}^{2\,x}\,\left (2\,{\mathrm {e}}^4-{\mathrm {e}}^2\,\left (4\,x^2+9\right )+9\,x^2+2\,x^4+9\right )+{\mathrm {e}}^x\,\left (20\,x^2-20\,{\mathrm {e}}^2+45\right )+50} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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