Optimal. Leaf size=17 \[ \frac {3}{2} x \left (2+\frac {2 (x+\log (\log (x)))}{e^3}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps
used = 10, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 6820,
6874, 2335, 2600} \begin {gather*} \frac {3 x^2}{e^3}+3 x+\frac {3 x \log (\log (x))}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2335
Rule 2600
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {\frac {6}{e^3}+\left (6+\frac {12 x}{e^3}\right ) \log (x)+\frac {6 \log (x) \log (\log (x))}{e^3}}{\log (x)} \, dx\\ &=\frac {1}{2} \int \frac {6+6 \log (x) \left (e^3+2 x+\log (\log (x))\right )}{e^3 \log (x)} \, dx\\ &=\frac {\int \frac {6+6 \log (x) \left (e^3+2 x+\log (\log (x))\right )}{\log (x)} \, dx}{2 e^3}\\ &=\frac {\int \left (\frac {6 \left (1+e^3 \log (x)+2 x \log (x)\right )}{\log (x)}+6 \log (\log (x))\right ) \, dx}{2 e^3}\\ &=\frac {3 \int \frac {1+e^3 \log (x)+2 x \log (x)}{\log (x)} \, dx}{e^3}+\frac {3 \int \log (\log (x)) \, dx}{e^3}\\ &=\frac {3 x \log (\log (x))}{e^3}+\frac {3 \int \left (e^3+2 x+\frac {1}{\log (x)}\right ) \, dx}{e^3}-\frac {3 \int \frac {1}{\log (x)} \, dx}{e^3}\\ &=3 x+\frac {3 x^2}{e^3}+\frac {3 x \log (\log (x))}{e^3}-\frac {3 \text {li}(x)}{e^3}+\frac {3 \int \frac {1}{\log (x)} \, dx}{e^3}\\ &=3 x+\frac {3 x^2}{e^3}+\frac {3 x \log (\log (x))}{e^3}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.29 \begin {gather*} \frac {3 e^3 x+3 x^2+3 x \log (\log (x))}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 20, normalized size = 1.18
method | result | size |
default | \(3 x +3 \,{\mathrm e}^{-3} x^{2}+3 \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{-3} x\) | \(20\) |
risch | \(3 x +3 \,{\mathrm e}^{-3} x^{2}+3 \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{-3} x\) | \(20\) |
norman | \(3 x +3 \,{\mathrm e}^{-3} x^{2}+3 \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{-3} x\) | \(24\) |
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 = 33, normalized size = 1.94 \begin {gather*} 3 \, x^{2} e^{\left (-3\right )} + 3 \, {\left (x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right )\right )} e^{\left (-3\right )} + 3 \, {\rm Ei}\left (\log \left (x\right )\right ) e^{\left (-3\right )} + 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 25, normalized size = 1.47 \begin {gather*} \frac {3}{2} \, x^{2} e^{\left (\log \left (2\right ) - 3\right )} + \frac {3}{2} \, x e^{\left (\log \left (2\right ) - 3\right )} \log \left (\log \left (x\right )\right ) + 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 22, normalized size = 1.29 \begin {gather*} \frac {3 x^{2}}{e^{3}} + \frac {3 x \log {\left (\log {\left (x \right )} \right )}}{e^{3}} + 3 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 19, normalized size = 1.12 \begin {gather*} 3 \, x^{2} e^{\left (-3\right )} + 3 \, x e^{\left (-3\right )} \log \left (\log \left (x\right )\right ) + 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.29, size = 12, normalized size = 0.71 \begin {gather*} 3\,x\,{\mathrm {e}}^{-3}\,\left (x+\ln \left (\ln \left (x\right )\right )+{\mathrm {e}}^3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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