Integrand size = 13, antiderivative size = 12 \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\frac {1}{2} \arctan \left (\frac {e^x}{2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2281, 209} \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\frac {1}{2} \arctan \left (\frac {e^x}{2}\right ) \]
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Rule 209
Rule 2281
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{4+x^2} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \tan ^{-1}\left (\frac {e^x}{2}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\frac {1}{2} \arctan \left (\frac {e^x}{2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {\arctan \left (\frac {{\mathrm e}^{x}}{2}\right )}{2}\) | \(8\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{x}+2 i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}-2 i\right )}{4}\) | \(20\) |
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none
Time = 0.30 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\frac {1}{2} \, \arctan \left (\frac {1}{2} \, e^{x}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\operatorname {RootSum} {\left (16 z^{2} + 1, \left ( i \mapsto i \log {\left (8 i + e^{x} \right )} \right )\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\frac {1}{2} \, \arctan \left (\frac {1}{2} \, e^{x}\right ) \]
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none
Time = 0.31 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\frac {1}{2} \, \arctan \left (\frac {1}{2} \, e^{x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {e^x}{4+e^{2 x}} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x}{2}\right )}{2} \]
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