Optimal. Leaf size=19 \[ \log \left (e^x-\frac {12 \log \left (\frac {\log (2)}{3}\right )}{\log (\log (x))}\right ) \]
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Rubi [F]
time = 2.39, 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 {12 \log \left (\frac {\log (2)}{3}\right )+e^x x \log (x) \log ^2(\log (x))}{-12 x \log (x) \log \left (\frac {\log (2)}{3}\right ) \log (\log (x))+e^x x \log (x) \log ^2(\log (x))} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-12 \log \left (\frac {\log (2)}{3}\right )-e^x x \log (x) \log ^2(\log (x))}{x \log (x) \log (\log (x)) \left (12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))\right )} \, dx\\ &=\int \left (1-\frac {12 \log \left (\frac {\log (2)}{3}\right ) (1+x \log (x) \log (\log (x)))}{x \log (x) \log (\log (x)) \left (12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))\right )}\right ) \, dx\\ &=x-\left (12 \log \left (\frac {\log (2)}{3}\right )\right ) \int \frac {1+x \log (x) \log (\log (x))}{x \log (x) \log (\log (x)) \left (12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))\right )} \, dx\\ &=x-\left (12 \log \left (\frac {\log (2)}{3}\right )\right ) \int \left (\frac {1}{12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))}-\frac {1}{x \log (x) \log (\log (x)) \left (-12 \log \left (\frac {\log (2)}{3}\right )+e^x \log (\log (x))\right )}\right ) \, dx\\ &=x-\left (12 \log \left (\frac {\log (2)}{3}\right )\right ) \int \frac {1}{12 \log \left (\frac {\log (2)}{3}\right )-e^x \log (\log (x))} \, dx+\left (12 \log \left (\frac {\log (2)}{3}\right )\right ) \int \frac {1}{x \log (x) \log (\log (x)) \left (-12 \log \left (\frac {\log (2)}{3}\right )+e^x \log (\log (x))\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 25, normalized size = 1.32 \begin {gather*} -\log (\log (\log (x)))+\log \left (-12 \log \left (\frac {\log (2)}{3}\right )+e^x \log (\log (x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 28, normalized size = 1.47
method | result | size |
risch | \(x -\ln \left (\ln \left (\ln \left (x \right )\right )\right )+\ln \left (\ln \left (\ln \left (x \right )\right )+12 \left (\ln \left (3\right )-\ln \left (\ln \left (2\right )\right )\right ) {\mathrm e}^{-x}\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 30, normalized size = 1.58 \begin {gather*} x + \log \left ({\left (e^{x} \log \left (\log \left (x\right )\right ) + 12 \, \log \left (3\right ) - 12 \, \log \left (\log \left (2\right )\right )\right )} e^{\left (-x\right )}\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 28, normalized size = 1.47 \begin {gather*} x + \log \left ({\left (e^{x} \log \left (\log \left (x\right )\right ) - 12 \, \log \left (\frac {1}{3} \, \log \left (2\right )\right )\right )} e^{\left (-x\right )}\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 20, normalized size = 1.05 \begin {gather*} \log {\left (e^{x} + \frac {- 12 \log {\left (\log {\left (2 \right )} \right )} + 12 \log {\left (3 \right )}}{\log {\left (\log {\left (x \right )} \right )}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 24, normalized size = 1.26 \begin {gather*} \log \left (e^{x} \log \left (\log \left (x\right )\right ) + 12 \, \log \left (3\right ) - 12 \, \log \left (\log \left (2\right )\right )\right ) - \log \left (\log \left (\log \left (x\right )\right )\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 = -0.05 \begin {gather*} \int -\frac {x\,{\mathrm {e}}^x\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+12\,\ln \left (\frac {\ln \left (2\right )}{3}\right )}{12\,x\,\ln \left (\frac {\ln \left (2\right )}{3}\right )\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )-x\,{\ln \left (\ln \left (x\right )\right )}^2\,{\mathrm {e}}^x\,\ln \left (x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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