Optimal. Leaf size=18 \[ \left (4+\frac {x^2}{\log (3) \log (6 x)}\right )^2 \]
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Rubi [A]
time = 0.23, antiderivative size = 30, normalized size of antiderivative = 1.67, number of steps
used = 5, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {12, 6873, 6844}
\begin {gather*} \frac {x^4}{\log ^2(3) \log ^2(6 x)}+\frac {8 x^2}{\log (3) \log (6 x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6844
Rule 6873
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-2 x^3+\left (4 x^3-8 x \log (3)\right ) \log (6 x)+16 x \log (3) \log ^2(6 x)}{\log ^3(6 x)} \, dx}{\log ^2(3)}\\ &=\frac {\int \frac {2 x (1-2 \log (6 x)) \left (-x^2-4 \log (3) \log (6 x)\right )}{\log ^3(6 x)} \, dx}{\log ^2(3)}\\ &=\frac {2 \int \frac {x (1-2 \log (6 x)) \left (-x^2-4 \log (3) \log (6 x)\right )}{\log ^3(6 x)} \, dx}{\log ^2(3)}\\ &=-\frac {2 \text {Subst}\left (\int (-x-4 \log (3)) \, dx,x,\frac {x^2}{\log (6 x)}\right )}{\log ^2(3)}\\ &=\frac {x^4}{\log ^2(3) \log ^2(6 x)}+\frac {8 x^2}{\log (3) \log (6 x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 33, normalized size = 1.83 \begin {gather*} \frac {2 \left (\frac {x^4}{2 \log ^2(6 x)}+\frac {4 x^2 \log (3)}{\log (6 x)}\right )}{\log ^2(3)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.36, size = 55, normalized size = 3.06
method | result | size |
risch | \(\frac {x^{2} \left (8 \ln \left (3\right ) \ln \left (6 x \right )+x^{2}\right )}{\ln \left (3\right )^{2} \ln \left (6 x \right )^{2}}\) | \(27\) |
norman | \(\frac {\frac {x^{4}}{\ln \left (3\right )}+8 \ln \left (6 x \right ) x^{2}}{\ln \left (3\right ) \ln \left (6 x \right )^{2}}\) | \(30\) |
default | \(\frac {-\frac {4 \ln \left (3\right ) \expIntegral \left (1, -2 \ln \left (6 x \right )\right )}{9}-\frac {2 \ln \left (3\right ) \left (-\frac {36 x^{2}}{\ln \left (6 x \right )}-2 \expIntegral \left (1, -2 \ln \left (6 x \right )\right )\right )}{9}+\frac {x^{4}}{\ln \left (6 x \right )^{2}}}{\ln \left (3\right )^{2}}\) | \(55\) |
derivativedivides | \(\frac {-288 \ln \left (3\right ) \expIntegral \left (1, -2 \ln \left (6 x \right )\right )-144 \ln \left (3\right ) \left (-\frac {36 x^{2}}{\ln \left (6 x \right )}-2 \expIntegral \left (1, -2 \ln \left (6 x \right )\right )\right )+\frac {648 x^{4}}{\ln \left (6 x \right )^{2}}}{648 \ln \left (3\right )^{2}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.31, size = 48, normalized size = 2.67 \begin {gather*} \frac {36 \, {\rm Ei}\left (2 \, \log \left (6 \, x\right )\right ) \log \left (3\right ) - 36 \, \Gamma \left (-1, -2 \, \log \left (6 \, x\right )\right ) \log \left (3\right ) + \Gamma \left (-1, -4 \, \log \left (6 \, x\right )\right ) + 2 \, \Gamma \left (-2, -4 \, \log \left (6 \, x\right )\right )}{81 \, \log \left (3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 26, normalized size = 1.44 \begin {gather*} \frac {x^{4} + 8 \, x^{2} \log \left (3\right ) \log \left (6 \, x\right )}{\log \left (3\right )^{2} \log \left (6 \, x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 27, normalized size = 1.50 \begin {gather*} \frac {x^{4} + 8 x^{2} \log {\left (3 \right )} \log {\left (6 x \right )}}{\log {\left (3 \right )}^{2} \log {\left (6 x \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 29, normalized size = 1.61 \begin {gather*} \frac {\frac {x^{4}}{\log \left (6 \, x\right )^{2}} + \frac {8 \, x^{2} \log \left (3\right )}{\log \left (6 \, x\right )}}{\log \left (3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.94, size = 26, normalized size = 1.44 \begin {gather*} \frac {x^2\,\left (8\,\ln \left (6\,x\right )\,\ln \left (3\right )+x^2\right )}{{\ln \left (6\,x\right )}^2\,{\ln \left (3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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