Integrand size = 17, antiderivative size = 18 \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=-\frac {x}{3}+\frac {2}{3} \log \left (3+e^{2 x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2320, 78} \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=\frac {2}{3} \log \left (e^{2 x}+3\right )-\frac {x}{3} \]
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Rule 78
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {-1+x}{x (3+x)} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{3 x}+\frac {4}{3 (3+x)}\right ) \, dx,x,e^{2 x}\right ) \\ & = -\frac {x}{3}+\frac {2}{3} \log \left (3+e^{2 x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=-\frac {1}{3} \log \left (e^x\right )+\frac {2}{3} \log \left (3+e^{2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78
method | result | size |
norman | \(-\frac {x}{3}+\frac {2 \ln \left (3+{\mathrm e}^{2 x}\right )}{3}\) | \(14\) |
risch | \(-\frac {x}{3}+\frac {2 \ln \left (3+{\mathrm e}^{2 x}\right )}{3}\) | \(14\) |
parallelrisch | \(-\frac {x}{3}+\frac {2 \ln \left (3+{\mathrm e}^{2 x}\right )}{3}\) | \(14\) |
derivativedivides | \(-\frac {\ln \left ({\mathrm e}^{2 x}\right )}{6}+\frac {2 \ln \left (3+{\mathrm e}^{2 x}\right )}{3}\) | \(18\) |
default | \(-\frac {\ln \left ({\mathrm e}^{2 x}\right )}{6}+\frac {2 \ln \left (3+{\mathrm e}^{2 x}\right )}{3}\) | \(18\) |
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Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=-\frac {1}{3} \, x + \frac {2}{3} \, \log \left (e^{\left (2 \, x\right )} + 3\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=- \frac {x}{3} + \frac {2 \log {\left (e^{2 x} + 3 \right )}}{3} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=-\frac {1}{3} \, x + \frac {2}{3} \, \log \left (e^{\left (2 \, x\right )} + 3\right ) \]
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Time = 0.31 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=-\frac {1}{3} \, x + \frac {2}{3} \, \log \left (e^{\left (2 \, x\right )} + 3\right ) \]
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Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {-1+e^{2 x}}{3+e^{2 x}} \, dx=\frac {2\,\ln \left ({\mathrm {e}}^{2\,x}+3\right )}{3}-\frac {x}{3} \]
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