Optimal. Leaf size=32 \[ \log \left (3+e^{\frac {e^{-x} \left (3+\frac {x+x^2}{\log \left (4 x^2\right )}\right )}{\log (\log (4))}}\right ) \]
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Rubi [F] time = 135.73, 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 {\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right ) \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+\exp \left (x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right ) \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-2 (1+x)+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{\left (1+3 \exp \left (-\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx\\ &=\frac {\int \frac {e^{-x} \left (-2 (1+x)+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{\left (1+3 \exp \left (-\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}\\ &=\frac {\int \left (\frac {e^{-x} \left (-2-2 x+\log \left (4 x^2\right )+x \log \left (4 x^2\right )-x^2 \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{\log ^2\left (4 x^2\right )}+\frac {3 e^{-x} \left (2+2 x-\log \left (4 x^2\right )-x \log \left (4 x^2\right )+x^2 \log \left (4 x^2\right )+3 \log ^2\left (4 x^2\right )\right )}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )}\right ) \, dx}{\log (\log (4))}\\ &=\frac {\int \frac {e^{-x} \left (-2-2 x+\log \left (4 x^2\right )+x \log \left (4 x^2\right )-x^2 \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{\log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {3 \int \frac {e^{-x} \left (2+2 x-\log \left (4 x^2\right )-x \log \left (4 x^2\right )+x^2 \log \left (4 x^2\right )+3 \log ^2\left (4 x^2\right )\right )}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}\\ &=\frac {\int \left (-3 e^{-x}-\frac {2 e^{-x} (1+x)}{\log ^2\left (4 x^2\right )}+\frac {e^{-x} \left (1+x-x^2\right )}{\log \left (4 x^2\right )}\right ) \, dx}{\log (\log (4))}+\frac {3 \int \frac {e^{-x} \left (2 (1+x)+\left (-1-x+x^2\right ) \log \left (4 x^2\right )+3 \log ^2\left (4 x^2\right )\right )}{\left (3+\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}\\ &=\frac {\int \frac {e^{-x} \left (1+x-x^2\right )}{\log \left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {2 \int \frac {e^{-x} (1+x)}{\log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {3 \int e^{-x} \, dx}{\log (\log (4))}+\frac {3 \int \left (\frac {3 e^{-x}}{3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )}+\frac {2 e^{-x}}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )}+\frac {2 e^{-x} x}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )}-\frac {e^{-x}}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )}-\frac {e^{-x} x}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )}+\frac {e^{-x} x^2}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )}\right ) \, dx}{\log (\log (4))}\\ &=\frac {3 e^{-x}}{\log (\log (4))}+\frac {\int \left (\frac {e^{-x}}{\log \left (4 x^2\right )}+\frac {e^{-x} x}{\log \left (4 x^2\right )}-\frac {e^{-x} x^2}{\log \left (4 x^2\right )}\right ) \, dx}{\log (\log (4))}-\frac {2 \int \left (\frac {e^{-x}}{\log ^2\left (4 x^2\right )}+\frac {e^{-x} x}{\log ^2\left (4 x^2\right )}\right ) \, dx}{\log (\log (4))}-\frac {3 \int \frac {e^{-x}}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {3 \int \frac {e^{-x} x}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {3 \int \frac {e^{-x} x^2}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {6 \int \frac {e^{-x}}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {6 \int \frac {e^{-x} x}{\left (3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {9 \int \frac {e^{-x}}{3+\exp \left (\frac {3 e^{-x}}{\log (\log (4))}+\frac {e^{-x} x}{\log \left (4 x^2\right ) \log (\log (4))}+\frac {e^{-x} x^2}{\log \left (4 x^2\right ) \log (\log (4))}\right )} \, dx}{\log (\log (4))}\\ &=\frac {3 e^{-x}}{\log (\log (4))}+\frac {\int \frac {e^{-x}}{\log \left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {\int \frac {e^{-x} x}{\log \left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {\int \frac {e^{-x} x^2}{\log \left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {2 \int \frac {e^{-x}}{\log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {2 \int \frac {e^{-x} x}{\log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {3 \int \frac {e^{-x}}{\left (3+\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )} \, dx}{\log (\log (4))}-\frac {3 \int \frac {e^{-x} x}{\left (3+\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {3 \int \frac {e^{-x} x^2}{\left (3+\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log \left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {6 \int \frac {e^{-x}}{\left (3+\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {6 \int \frac {e^{-x} x}{\left (3+\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )\right ) \log ^2\left (4 x^2\right )} \, dx}{\log (\log (4))}+\frac {9 \int \frac {e^{-x}}{3+\exp \left (\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}\right )} \, dx}{\log (\log (4))}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 1.20, size = 82, normalized size = 2.56 \begin {gather*} \frac {3 e^{-x}}{\log (\log (4))}+\frac {\frac {e^{-x} x (1+x)}{\log \left (4 x^2\right )}+\log \left (1+3 e^{-\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}}\right ) \log (\log (4))}{\log (\log (4))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 59, normalized size = 1.84 \begin {gather*} -x + \log \left (3 \, e^{x} + e^{\left (\frac {{\left (x e^{x} \log \left (4 \, x^{2}\right ) \log \left (2 \, \log \relax (2)\right ) + x^{2} + x + 3 \, \log \left (4 \, x^{2}\right )\right )} e^{\left (-x\right )}}{\log \left (4 \, x^{2}\right ) \log \left (2 \, \log \relax (2)\right )}\right )}\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*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.01, size = 329, normalized size = 10.28
method | result | size |
risch | \(\frac {3 \,{\mathrm e}^{-x}}{\ln \relax (2)+\ln \left (\ln \relax (2)\right )}+\frac {2 i \left (x +1\right ) x \,{\mathrm e}^{-x}}{\left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (2)+4 i \ln \relax (x )\right ) \left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right )}-\frac {\left (6 \ln \relax (2)+6 \ln \relax (x )-\frac {3 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}+x^{2}+x \right ) {\mathrm e}^{-x}}{\left (2 \ln \relax (2)+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right ) \left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right )}+\ln \left ({\mathrm e}^{\frac {\left (-3 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+6 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-3 i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 x^{2}+12 \ln \relax (2)+12 \ln \relax (x )+2 x \right ) {\mathrm e}^{-x}}{\left (-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+4 \ln \relax (2)+4 \ln \relax (x )\right ) \left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right )}}+3\right )\) | \(329\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 170, normalized size = 5.31 \begin {gather*} \frac {{\left (x + 6 \, \log \relax (2) + 6 \, \log \relax (x)\right )} e^{\left (-x\right )}}{2 \, {\left (\log \relax (2)^{2} + {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \log \relax (x) + \log \relax (2) \log \left (\log \relax (2)\right )\right )}} + \log \left ({\left (e^{\left (\frac {x^{2}}{2 \, {\left ({\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} e^{x} \log \relax (x) + {\left (\log \relax (2)^{2} + \log \relax (2) \log \left (\log \relax (2)\right )\right )} e^{x}\right )}} + \frac {x}{2 \, {\left ({\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} e^{x} \log \relax (x) + {\left (\log \relax (2)^{2} + \log \relax (2) \log \left (\log \relax (2)\right )\right )} e^{x}\right )}} + \frac {3 \, e^{\left (-x\right )}}{\log \relax (2) + \log \left (\log \relax (2)\right )}\right )} + 3\right )} e^{\left (-\frac {x}{2 \, {\left ({\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} e^{x} \log \relax (x) + {\left (\log \relax (2)^{2} + \log \relax (2) \log \left (\log \relax (2)\right )\right )} e^{x}\right )}} - \frac {3 \, e^{\left (-x\right )}}{\log \relax (2) + \log \left (\log \relax (2)\right )}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 107, normalized size = 3.34 \begin {gather*} \ln \left ({\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,\ln \left (64\,x^6\right )}{\ln \relax (4)\,\ln \left (\ln \relax (2)\right )+2\,{\ln \relax (2)}^2+\ln \left (2\,\ln \relax (2)\right )\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{-x}}{\ln \relax (4)\,\ln \left (\ln \relax (2)\right )+2\,{\ln \relax (2)}^2+\ln \left (2\,\ln \relax (2)\right )\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{-x}}{\ln \relax (4)\,\ln \left (\ln \relax (2)\right )+2\,{\ln \relax (2)}^2+\ln \left (2\,\ln \relax (2)\right )\,\ln \left (x^2\right )}}+3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 34, normalized size = 1.06 \begin {gather*} \log {\left (e^{\frac {\left (x^{2} + x + 3 \log {\left (4 x^{2} \right )}\right ) e^{- x}}{\log {\left (4 x^{2} \right )} \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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