Optimal. Leaf size=27 \[ 5-\log ^2\left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right ) \]
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Rubi [F]
time = 1.39, 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 {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\left (2 x-x^2\right ) \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )+x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\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 {\left (-2+(4+2 x) \log (4 x)+2 x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right ) \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (2-x+\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )} \, dx\\ &=\int \left (-\frac {2 \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )}-\frac {2 \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}-\frac {4 \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{x \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}+\frac {2 \log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {\log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )} \, dx\right )-2 \int \frac {\log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )} \, dx+2 \int \frac {\log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{x \log (4 x) \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )} \, dx-4 \int \frac {\log \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )}{x \log \left (\frac {e^{-x} \log (4 x)}{x^2}\right ) \left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 25, normalized size = 0.93 \begin {gather*} -\log ^2\left (-2+x-\log \left (\log \left (\frac {e^{-x} \log (4 x)}{x^2}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x \ln \left (4 x \right ) \ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )+\left (2 x +4\right ) \ln \left (4 x \right )-2\right ) \ln \left (-\ln \left (\ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )\right )+x -2\right )}{x \ln \left (4 x \right ) \ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right ) \ln \left (\ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )\right )+\left (-x^{2}+2 x \right ) \ln \left (4 x \right ) \ln \left (\frac {\ln \left (4 x \right ) {\mathrm e}^{-x}}{x^{2}}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (26) = 52\).
time = 0.63, size = 71, normalized size = 2.63 \begin {gather*} -2 \, \log \left (x - \log \left (\log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right )\right ) - 2\right ) \log \left (-x + \log \left (-x - 2 \, \log \left (x\right ) + \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )\right ) + 2\right ) + \log \left (-x + \log \left (-x - 2 \, \log \left (x\right ) + \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )\right ) + 2\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 24, normalized size = 0.89 \begin {gather*} -\log \left (x - \log \left (\log \left (\frac {e^{\left (-x\right )} \log \left (4 \, x\right )}{x^{2}}\right )\right ) - 2\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.49, size = 24, normalized size = 0.89 \begin {gather*} -{\ln \left (x-\ln \left (\ln \left (\frac {\ln \left (4\,x\right )\,{\mathrm {e}}^{-x}}{x^2}\right )\right )-2\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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