Optimal. Leaf size=20 \[ x+2 \left (x+\frac {\frac {1}{e}-x}{x+\log (4)}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 12, 1850} \begin {gather*} 3 x+\frac {2 (1+e \log (4))}{e (x+\log (4))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 1850
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{e (x+\log (4))^2} \, dx\\ &=\frac {\int \frac {-2+3 e x^2+e (-2+6 x) \log (4)+3 e \log ^2(4)}{(x+\log (4))^2} \, dx}{e}\\ &=\frac {\int \left (3 e-\frac {2 (1+e \log (4))}{(x+\log (4))^2}\right ) \, dx}{e}\\ &=3 x+\frac {2 (1+e \log (4))}{e (x+\log (4))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 25, normalized size = 1.25 \begin {gather*} \frac {3 e (x+\log (4))+\frac {2+e \log (16)}{x+\log (4)}}{e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 34, normalized size = 1.70 \begin {gather*} \frac {3 \, x^{2} e + 2 \, {\left (3 \, x + 2\right )} e \log \relax (2) + 2}{x e + 2 \, e \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 24, normalized size = 1.20 \begin {gather*} 3 \, x + \frac {2 \, {\left (2 \, e \log \relax (2) + 1\right )} e^{\left (-1\right )}}{x + 2 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 30, normalized size = 1.50
method | result | size |
default | \({\mathrm e}^{-1} \left (3 x \,{\mathrm e}-\frac {-4 \,{\mathrm e} \ln \relax (2)-2}{x +2 \ln \relax (2)}\right )\) | \(30\) |
risch | \(3 x +\frac {4 \,{\mathrm e}^{-1} {\mathrm e} \ln \relax (2)}{x +2 \ln \relax (2)}+\frac {2 \,{\mathrm e}^{-1}}{x +2 \ln \relax (2)}\) | \(33\) |
gosper | \(-\frac {\left (12 \,{\mathrm e} \ln \relax (2)^{2}-3 x^{2} {\mathrm e}-4 \,{\mathrm e} \ln \relax (2)-2\right ) {\mathrm e}^{-1}}{x +2 \ln \relax (2)}\) | \(38\) |
norman | \(\frac {3 x^{2}-2 \left (6 \,{\mathrm e} \ln \relax (2)^{2}-2 \,{\mathrm e} \ln \relax (2)-1\right ) {\mathrm e}^{-1}}{x +2 \ln \relax (2)}\) | \(38\) |
meijerg | \(\frac {3 x}{1+\frac {x}{2 \ln \relax (2)}}-\frac {{\mathrm e}^{-1} x}{2 \ln \relax (2)^{2} \left (1+\frac {x}{2 \ln \relax (2)}\right )}-\frac {x}{\ln \relax (2) \left (1+\frac {x}{2 \ln \relax (2)}\right )}+12 \ln \relax (2) \left (-\frac {x}{2 \ln \relax (2) \left (1+\frac {x}{2 \ln \relax (2)}\right )}+\ln \left (1+\frac {x}{2 \ln \relax (2)}\right )\right )+6 \ln \relax (2) \left (\frac {x \left (\frac {3 x}{2 \ln \relax (2)}+6\right )}{6 \ln \relax (2) \left (1+\frac {x}{2 \ln \relax (2)}\right )}-2 \ln \left (1+\frac {x}{2 \ln \relax (2)}\right )\right )\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 27, normalized size = 1.35 \begin {gather*} 3 \, x + \frac {2 \, {\left (2 \, e \log \relax (2) + 1\right )}}{x e + 2 \, e \log \relax (2)} \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 = -0.05 \begin {gather*} \int \frac {12\,\mathrm {e}\,{\ln \relax (2)}^2+3\,x^2\,\mathrm {e}+2\,\mathrm {e}\,\ln \relax (2)\,\left (6\,x-2\right )-2}{\mathrm {e}\,x^2+4\,\mathrm {e}\,\ln \relax (2)\,x+4\,\mathrm {e}\,{\ln \relax (2)}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 26, normalized size = 1.30 \begin {gather*} 3 x + \frac {2 + 4 e \log {\relax (2 )}}{e x + 2 e \log {\relax (2 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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