Optimal. Leaf size=10 \[ \frac {\log (\log (4))}{\log ^2(-x)} \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 2302, 30} \begin {gather*} \frac {\log (\log (4))}{\log ^2(-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2302
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left ((2 \log (\log (4))) \int \frac {1}{x \log ^3(-x)} \, dx\right )\\ &=-\left ((2 \log (\log (4))) \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (-x)\right )\right )\\ &=\frac {\log (\log (4))}{\log ^2(-x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log (\log (4))}{\log ^2(-x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 12, normalized size = 1.20 \begin {gather*} \frac {\log \left (2 \, \log \relax (2)\right )}{\log \left (-x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 12, normalized size = 1.20 \begin {gather*} \frac {\log \left (2 \, \log \relax (2)\right )}{\log \left (-x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 13, normalized size = 1.30
method | result | size |
derivativedivides | \(\frac {\ln \left (2 \ln \relax (2)\right )}{\ln \left (-x \right )^{2}}\) | \(13\) |
default | \(\frac {\ln \left (2 \ln \relax (2)\right )}{\ln \left (-x \right )^{2}}\) | \(13\) |
norman | \(\frac {\ln \relax (2)+\ln \left (\ln \relax (2)\right )}{\ln \left (-x \right )^{2}}\) | \(14\) |
risch | \(\frac {\ln \relax (2)}{\ln \left (-x \right )^{2}}+\frac {\ln \left (\ln \relax (2)\right )}{\ln \left (-x \right )^{2}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 12, normalized size = 1.20 \begin {gather*} \frac {\log \left (2 \, \log \relax (2)\right )}{\log \left (-x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 10, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\ln \relax (4)\right )}{{\ln \left (-x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 14, normalized size = 1.40 \begin {gather*} \frac {\log {\left (\log {\relax (2 )} \right )} + \log {\relax (2 )}}{\log {\left (- x \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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