Optimal. Leaf size=23 \[ 5+3 x+\frac {2 \left (1+e^3\right )}{1+x-\log (3)}+\log (4) \]
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Rubi [A]
time = 0.04, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps
used = 4, number of rules used = 3, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {2009, 27, 697}
\begin {gather*} 3 x+\frac {2 \left (1+e^3\right )}{x+1-\log (3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 697
Rule 2009
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1-2 e^3+3 x^2+6 x (1-\log (3))-6 \log (3)+3 \log ^2(3)}{x^2+2 x (1-\log (3))+(-1+\log (3))^2} \, dx\\ &=\int \frac {1-2 e^3+3 x^2+6 x (1-\log (3))-6 \log (3)+3 \log ^2(3)}{(1+x-\log (3))^2} \, dx\\ &=\int \left (3-\frac {2 \left (1+e^3\right )}{(1+x-\log (3))^2}\right ) \, dx\\ &=3 x+\frac {2 \left (1+e^3\right )}{1+x-\log (3)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 26, normalized size = 1.13 \begin {gather*} \frac {2 \left (1+e^3\right )}{1+x-\log (3)}+3 (1+x-\log (3)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 22, normalized size = 0.96
method | result | size |
default | \(3 x -\frac {-2-2 \,{\mathrm e}^{3}}{1+x -\ln \left (3\right )}\) | \(22\) |
risch | \(3 x -\frac {2}{\ln \left (3\right )-x -1}-\frac {2 \,{\mathrm e}^{3}}{\ln \left (3\right )-x -1}\) | \(29\) |
norman | \(\frac {-3 x^{2}+1+3 \ln \left (3\right )^{2}-2 \,{\mathrm e}^{3}-6 \ln \left (3\right )}{\ln \left (3\right )-x -1}\) | \(32\) |
gosper | \(-\frac {3 x^{2}-1-3 \ln \left (3\right )^{2}+2 \,{\mathrm e}^{3}+6 \ln \left (3\right )}{\ln \left (3\right )-x -1}\) | \(33\) |
meijerg | \(\left (-6 \ln \left (3\right )+6\right ) \left (\frac {x}{\left (\ln \left (3\right )-1\right ) \left (1-\frac {x}{\ln \left (3\right )-1}\right )}+\ln \left (1-\frac {x}{\ln \left (3\right )-1}\right )\right )+\frac {3 \left (\ln \left (3\right )-1\right )^{2} \left (-\frac {x \left (-\frac {3 x}{\ln \left (3\right )-1}+6\right )}{3 \left (\ln \left (3\right )-1\right ) \left (1-\frac {x}{\ln \left (3\right )-1}\right )}-2 \ln \left (1-\frac {x}{\ln \left (3\right )-1}\right )\right )}{1-\ln \left (3\right )}-\frac {3 \ln \left (3\right )^{2} x}{\left (1-\ln \left (3\right )\right ) \left (\ln \left (3\right )-1\right ) \left (1-\frac {x}{\ln \left (3\right )-1}\right )}+\frac {2 \,{\mathrm e}^{3} x}{\left (1-\ln \left (3\right )\right ) \left (\ln \left (3\right )-1\right ) \left (1-\frac {x}{\ln \left (3\right )-1}\right )}+\frac {6 \ln \left (3\right ) x}{\left (1-\ln \left (3\right )\right ) \left (\ln \left (3\right )-1\right ) \left (1-\frac {x}{\ln \left (3\right )-1}\right )}-\frac {x}{\left (1-\ln \left (3\right )\right ) \left (\ln \left (3\right )-1\right ) \left (1-\frac {x}{\ln \left (3\right )-1}\right )}\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 0.83 \begin {gather*} 3 \, x + \frac {2 \, {\left (e^{3} + 1\right )}}{x - \log \left (3\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 29, normalized size = 1.26 \begin {gather*} \frac {3 \, x^{2} - 3 \, x \log \left (3\right ) + 3 \, x + 2 \, e^{3} + 2}{x - \log \left (3\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 15, normalized size = 0.65 \begin {gather*} 3 x + \frac {2 + 2 e^{3}}{x - \log {\left (3 \right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 19, normalized size = 0.83 \begin {gather*} 3 \, x + \frac {2 \, {\left (e^{3} + 1\right )}}{x - \log \left (3\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 61, normalized size = 2.65 \begin {gather*} 3\,x-\frac {\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}-\ln \left (9\right )\,1{}\mathrm {i}+2{}\mathrm {i}}{\sqrt {2\,\ln \left (3\right )+\ln \left (9\right )}\,\sqrt {\ln \left (9\right )-2\,\ln \left (3\right )}}\right )\,\left ({\mathrm {e}}^3+1\right )\,4{}\mathrm {i}}{\sqrt {2\,\ln \left (3\right )+\ln \left (9\right )}\,\sqrt {\ln \left (9\right )-2\,\ln \left (3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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