Integrand size = 20, antiderivative size = 17 \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=-\log \left (1+e^x\right )+2 \log \left (2+e^x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2320, 646, 31} \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]
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Rule 31
Rule 646
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{2+3 x+x^2} \, dx,x,e^x\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right ) \\ & = -\log \left (1+e^x\right )+2 \log \left (2+e^x\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=-\log \left (1+e^x\right )+2 \log \left (2+e^x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\ln \left (1+{\mathrm e}^{x}\right )+2 \ln \left (2+{\mathrm e}^{x}\right )\) | \(16\) |
norman | \(-\ln \left (1+{\mathrm e}^{x}\right )+2 \ln \left (2+{\mathrm e}^{x}\right )\) | \(16\) |
risch | \(-\ln \left (1+{\mathrm e}^{x}\right )+2 \ln \left (2+{\mathrm e}^{x}\right )\) | \(16\) |
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none
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=- \log {\left (e^{x} + 1 \right )} + 2 \log {\left (e^{x} + 2 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
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Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {e^{2 x}}{2+3 e^x+e^{2 x}} \, dx=2\,\ln \left ({\mathrm {e}}^x+2\right )-\ln \left ({\mathrm {e}}^x+1\right ) \]
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