Integrand size = 15, antiderivative size = 10 \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\frac {1}{2} \arctan \left (e^{2 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2281, 209} \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\frac {1}{2} \arctan \left (e^{2 x}\right ) \]
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Rule 209
Rule 2281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{2} \tan ^{-1}\left (e^{2 x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\frac {1}{2} \arctan \left (e^{2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\arctan \left ({\mathrm e}^{2 x}\right )}{2}\) | \(8\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{2 x}+i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{2 x}-i\right )}{4}\) | \(24\) |
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none
Time = 0.29 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (7) = 14\).
Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.70 \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\operatorname {RootSum} {\left (16 z^{2} + 1, \left ( i \mapsto i \log {\left (4 i + e^{2 x} \right )} \right )\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\frac {1}{2} \, \arctan \left (e^{\left (2 \, x\right )}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {e^{2 x}}{1+e^{4 x}} \, dx=\frac {\mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right )}{2} \]
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