Optimal. Leaf size=9 \[ \log \left (\log \left (-e^2 \log (x)\right )\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2302, 29} \begin {gather*} \log \left (\log \left (-e^2 \log (x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2302
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {1}{x \log \left (-e^2 x\right )} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (-e^2 \log (x)\right )\right )\\ &=\log \left (\log \left (-e^2 \log (x)\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 8, normalized size = 0.89 \begin {gather*} \log (2+\log (-\log (x))) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 8, normalized size = 0.89 \begin {gather*} \log \left (\log \left (-e^{2} \log \relax (x)\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 9, normalized size = 1.00 \begin {gather*} \log \left ({\left | \log \left (-e^{2} \log \relax (x)\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 9, normalized size = 1.00
method | result | size |
derivativedivides | \(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\) | \(9\) |
default | \(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\) | \(9\) |
norman | \(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\) | \(9\) |
risch | \(\ln \left (\ln \left (-{\mathrm e}^{2} \ln \relax (x )\right )\right )\) | \(9\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 8, normalized size = 0.89 \begin {gather*} \log \left (\log \left (-e^{2} \log \relax (x)\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 8, normalized size = 0.89 \begin {gather*} \ln \left (\ln \left (-\ln \relax (x)\right )+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 10, normalized size = 1.11 \begin {gather*} \log {\left (\log {\left (- e^{2} \log {\relax (x )} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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