Optimal. Leaf size=24 \[ \frac {3}{x \left (3 x+\frac {1}{9} \log ^2\left (\log \left ((2+x)^2\right )\right )\right )} \]
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Rubi [A] time = 0.37, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 3, integrand size = 149, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6688, 12, 6687} \begin {gather*} \frac {27}{x \left (27 x+\log ^2\left (\log \left ((x+2)^2\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6687
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27 \left (-4 x \log \left (\log \left ((2+x)^2\right )\right )-(2+x) \log \left ((2+x)^2\right ) \left (54 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )\right )}{x^2 (2+x) \log \left ((2+x)^2\right ) \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )^2} \, dx\\ &=27 \int \frac {-4 x \log \left (\log \left ((2+x)^2\right )\right )-(2+x) \log \left ((2+x)^2\right ) \left (54 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )}{x^2 (2+x) \log \left ((2+x)^2\right ) \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )^2} \, dx\\ &=\frac {27}{x \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.42, size = 20, normalized size = 0.83 \begin {gather*} \frac {27}{x \left (27 x+\log ^2\left (\log \left ((2+x)^2\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 24, normalized size = 1.00 \begin {gather*} \frac {27}{x \log \left (\log \left (x^{2} + 4 \, x + 4\right )\right )^{2} + 27 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.95, size = 24, normalized size = 1.00 \begin {gather*} \frac {27}{x \log \left (\log \left (x^{2} + 4 \, x + 4\right )\right )^{2} + 27 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-27 x -54\right ) \ln \left (x^{2}+4 x +4\right ) \ln \left (\ln \left (x^{2}+4 x +4\right )\right )^{2}-108 x \ln \left (\ln \left (x^{2}+4 x +4\right )\right )+\left (-1458 x^{2}-2916 x \right ) \ln \left (x^{2}+4 x +4\right )}{\left (x^{3}+2 x^{2}\right ) \ln \left (x^{2}+4 x +4\right ) \ln \left (\ln \left (x^{2}+4 x +4\right )\right )^{4}+\left (54 x^{4}+108 x^{3}\right ) \ln \left (x^{2}+4 x +4\right ) \ln \left (\ln \left (x^{2}+4 x +4\right )\right )^{2}+\left (729 x^{5}+1458 x^{4}\right ) \ln \left (x^{2}+4 x +4\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 4.00, size = 35, normalized size = 1.46 \begin {gather*} \frac {27}{x \log \relax (2)^{2} + 2 \, x \log \relax (2) \log \left (\log \left (x + 2\right )\right ) + x \log \left (\log \left (x + 2\right )\right )^{2} + 27 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.75, size = 199, normalized size = 8.29 \begin {gather*} \frac {27\,{\left (2\,\ln \left (x^2+4\,x+4\right )+x\,\ln \left (x^2+4\,x+4\right )\right )}^2\,\left (27\,x^2\,{\ln \left (x^2+4\,x+4\right )}^2+108\,x\,{\ln \left (x^2+4\,x+4\right )}^2+16\,x+108\,{\ln \left (x^2+4\,x+4\right )}^2\right )}{x\,\ln \left (x^2+4\,x+4\right )\,\left ({\ln \left (\ln \left (x^2+4\,x+4\right )\right )}^2+27\,x\right )\,\left (x+2\right )\,\left (27\,x^3\,{\ln \left (x^2+4\,x+4\right )}^3+162\,x^2\,{\ln \left (x^2+4\,x+4\right )}^3+16\,x^2\,\ln \left (x^2+4\,x+4\right )+324\,x\,{\ln \left (x^2+4\,x+4\right )}^3+32\,x\,\ln \left (x^2+4\,x+4\right )+216\,{\ln \left (x^2+4\,x+4\right )}^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 20, normalized size = 0.83 \begin {gather*} \frac {27}{27 x^{2} + x \log {\left (\log {\left (x^{2} + 4 x + 4 \right )} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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