Optimal. Leaf size=24 \[ x+e^{2-x \left (1+\frac {4}{\log ^2(3)}\right )} \log (4+2 x) \]
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Rubi [A]
time = 0.35, antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps
used = 4, number of rules used = 3, integrand size = 88, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {12, 6874, 2326}
\begin {gather*} x+\frac {e^{2-x \left (1+\frac {4}{\log ^2(3)}\right )} (x \log (2 (x+2))+2 \log (2 (x+2)))}{x+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{\frac {-4 x+(2-x) \log ^2(3)}{\log ^2(3)}} \log ^2(3)+(2+x) \log ^2(3)+e^{\frac {-4 x+(2-x) \log ^2(3)}{\log ^2(3)}} \left (-8-4 x+(-2-x) \log ^2(3)\right ) \log (4+2 x)}{2+x} \, dx}{\log ^2(3)}\\ &=\frac {\int \left (\log ^2(3)+\frac {e^{2-x \left (1+\frac {4}{\log ^2(3)}\right )} \left (\log ^2(3)-8 \left (1+\frac {\log ^2(3)}{4}\right ) \log (2 (2+x))-4 x \left (1+\frac {\log ^2(3)}{4}\right ) \log (2 (2+x))\right )}{2+x}\right ) \, dx}{\log ^2(3)}\\ &=x+\frac {\int \frac {e^{2-x \left (1+\frac {4}{\log ^2(3)}\right )} \left (\log ^2(3)-8 \left (1+\frac {\log ^2(3)}{4}\right ) \log (2 (2+x))-4 x \left (1+\frac {\log ^2(3)}{4}\right ) \log (2 (2+x))\right )}{2+x} \, dx}{\log ^2(3)}\\ &=x+\frac {e^{2-x \left (1+\frac {4}{\log ^2(3)}\right )} (2 \log (2 (2+x))+x \log (2 (2+x)))}{2+x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.80, size = 24, normalized size = 1.00 \begin {gather*} 2+x+e^{2-x-\frac {4 x}{\log ^2(3)}} \log (2 (2+x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.21, size = 44, normalized size = 1.83
method | result | size |
risch | \(x +{\mathrm e}^{-\frac {x \ln \left (3\right )^{2}-2 \ln \left (3\right )^{2}+4 x}{\ln \left (3\right )^{2}}} \ln \left (2 x +4\right )\) | \(33\) |
norman | \(\frac {x \ln \left (3\right )+\ln \left (3\right ) {\mathrm e}^{\frac {\left (2-x \right ) \ln \left (3\right )^{2}-4 x}{\ln \left (3\right )^{2}}} \ln \left (2 x +4\right )}{\ln \left (3\right )}\) | \(40\) |
default | \(\frac {x \ln \left (3\right )^{2}+\ln \left (3\right )^{2} {\mathrm e}^{\frac {\left (2-x \right ) \ln \left (3\right )^{2}-4 x}{\ln \left (3\right )^{2}}} \ln \left (2 x +4\right )}{\ln \left (3\right )^{2}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (-\frac {{\left (x - 2\right )} \log \left (3\right )^{2} + 4 \, x}{\log \left (3\right )^{2}}\right )} \log \left (2 \, x + 4\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 26, normalized size = 1.08 \begin {gather*} x + e^{\frac {- 4 x + \left (2 - x\right ) \log {\left (3 \right )}^{2}}{\log {\left (3 \right )}^{2}}} \log {\left (2 x + 4 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\ln \left (3\right )}^2\,\left (x+2\right )+{\mathrm {e}}^{-\frac {4\,x+{\ln \left (3\right )}^2\,\left (x-2\right )}{{\ln \left (3\right )}^2}}\,{\ln \left (3\right )}^2-{\mathrm {e}}^{-\frac {4\,x+{\ln \left (3\right )}^2\,\left (x-2\right )}{{\ln \left (3\right )}^2}}\,\ln \left (2\,x+4\right )\,\left (4\,x+{\ln \left (3\right )}^2\,\left (x+2\right )+8\right )}{{\ln \left (3\right )}^2\,\left (x+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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