Optimal. Leaf size=30 \[ e^{\frac {\frac {1}{2}-e^x}{-i \pi +x-\log (4-\log (2))}} \]
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Rubi [F] time = 5.06, 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 {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right ) \left (-1+e^x (2-2 x)+2 e^x (i \pi +\log (4-\log (2)))\right )}{2 x^2-4 x (i \pi +\log (4-\log (2)))+2 (i \pi +\log (4-\log (2)))^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int -\frac {\exp \left (\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right ) \left (-1+e^x (2-2 x)+2 e^x (i \pi +\log (4-\log (2)))\right )}{2 (\pi +i x-i \log (4-\log (2)))^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\exp \left (\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right ) \left (-1+e^x (2-2 x)+2 e^x (i \pi +\log (4-\log (2)))\right )}{(\pi +i x-i \log (4-\log (2)))^2} \, dx\right )\\ &=-\left (\frac {1}{2} \int \left (-\frac {\exp \left (\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{(\pi +i x-i \log (4-\log (2)))^2}+\frac {2 \exp \left (x+\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right ) (1+i \pi -x+\log (4-\log (2)))}{(\pi +i x-i \log (4-\log (2)))^2}\right ) \, dx\right )\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{(\pi +i x-i \log (4-\log (2)))^2} \, dx-\int \frac {\exp \left (x+\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right ) (1+i \pi -x+\log (4-\log (2)))}{(\pi +i x-i \log (4-\log (2)))^2} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{(\pi +i x-i \log (4-\log (2)))^2} \, dx-\int \left (\frac {\exp \left (x+\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{(\pi +i x-i \log (4-\log (2)))^2}+\frac {i \exp \left (x+\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{\pi +i x-i \log (4-\log (2))}\right ) \, dx\\ &=-\left (i \int \frac {\exp \left (x+\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{\pi +i x-i \log (4-\log (2))} \, dx\right )+\frac {1}{2} \int \frac {\exp \left (\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{(\pi +i x-i \log (4-\log (2)))^2} \, dx-\int \frac {\exp \left (x+\frac {-1+2 e^x}{-2 x+2 (i \pi +\log (4-\log (2)))}\right )}{(\pi +i x-i \log (4-\log (2)))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.50, size = 31, normalized size = 1.03 \begin {gather*} e^{\frac {-1+2 e^x}{2 (i \pi -x+\log (4-\log (2)))}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 20, normalized size = 0.67 \begin {gather*} e^{\left (-\frac {2 \, e^{x} - 1}{2 \, {\left (x - \log \left (\log \relax (2) - 4\right )\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 30, normalized size = 1.00 \begin {gather*} e^{\left (-\frac {e^{x}}{x - \log \left (\log \relax (2) - 4\right )} + \frac {1}{2 \, {\left (x - \log \left (\log \relax (2) - 4\right )\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 21, normalized size = 0.70
method | result | size |
risch | \({\mathrm e}^{-\frac {2 \,{\mathrm e}^{x}-1}{2 \left (x -\ln \left (\ln \relax (2)-4\right )\right )}}\) | \(21\) |
norman | \(\frac {\ln \left (\ln \relax (2)-4\right ) {\mathrm e}^{\frac {2 \,{\mathrm e}^{x}-1}{2 \ln \left (\ln \relax (2)-4\right )-2 x}}-x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{x}-1}{2 \ln \left (\ln \relax (2)-4\right )-2 x}}}{\ln \left (\ln \relax (2)-4\right )-x}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 30, normalized size = 1.00 \begin {gather*} e^{\left (-\frac {e^{x}}{x - \log \left (\log \relax (2) - 4\right )} + \frac {1}{2 \, {\left (x - \log \left (\log \relax (2) - 4\right )\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 31, normalized size = 1.03 \begin {gather*} {\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{x-\ln \left (\ln \relax (2)-4\right )}}\,{\mathrm {e}}^{\frac {1}{2\,x-2\,\ln \left (\ln \relax (2)-4\right )}} \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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