Integrand size = 21, antiderivative size = 28 \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {4+\frac {e^{2 x}}{2}}{4 \left (3+e^{\frac {1}{3+e^2}}\right )} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 2225} \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {e^{2 x}}{8 \left (3+e^{\frac {1}{3+e^2}}\right )} \]
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Rule 12
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {\int e^{2 x} \, dx}{4 \left (3+e^{\frac {1}{3+e^2}}\right )} \\ & = \frac {e^{2 x}}{8 \left (3+e^{\frac {1}{3+e^2}}\right )} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {e^{2 x}}{2 \left (12+4 e^{\frac {1}{3+e^2}}\right )} \]
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Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {{\mathrm e}^{2 x}}{8 \,{\mathrm e}^{\frac {1}{{\mathrm e}^{2}+3}}+24}\) | \(18\) |
default | \(\frac {{\mathrm e}^{2 x}}{8 \,{\mathrm e}^{\frac {1}{{\mathrm e}^{2}+3}}+24}\) | \(18\) |
norman | \(\frac {{\mathrm e}^{2 x}}{8 \,{\mathrm e}^{\frac {1}{{\mathrm e}^{2}+3}}+24}\) | \(18\) |
derivativedivides | \(\frac {{\mathrm e}^{2 x}}{8 \,{\mathrm e}^{\frac {1}{{\mathrm e}^{2}+3}}+24}\) | \(20\) |
risch | \(\frac {{\mathrm e}^{2 x}}{8 \,{\mathrm e}^{\frac {1}{{\mathrm e}^{2}+3}}+24}\) | \(20\) |
parallelrisch | \(\frac {{\mathrm e}^{2 x}}{8 \,{\mathrm e}^{\frac {1}{{\mathrm e}^{2}+3}}+24}\) | \(20\) |
meijerg | \(-\frac {1-{\mathrm e}^{2 x}}{2 \left (4 \,{\mathrm e}^{\frac {1}{{\mathrm e}^{2}+3}}+12\right )}\) | \(24\) |
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none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {e^{\left (2 \, x\right )}}{8 \, {\left (e^{\left (\frac {1}{e^{2} + 3}\right )} + 3\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {e^{2 x}}{8 e^{\frac {1}{3 + e^{2}}} + 24} \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {e^{\left (2 \, x\right )}}{8 \, {\left (e^{\left (\frac {1}{e^{2} + 3}\right )} + 3\right )}} \]
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {e^{\left (2 \, x\right )}}{8 \, {\left (e^{\left (\frac {1}{e^{2} + 3}\right )} + 3\right )}} \]
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Time = 8.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64 \[ \int \frac {e^{2 x}}{12+4 e^{\frac {1}{3+e^2}}} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8\,{\mathrm {e}}^{\frac {1}{{\mathrm {e}}^2+3}}+24} \]
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