Optimal. Leaf size=9 \[ -\frac{1}{e^x+1} \]
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Rubi [A] time = 0.0241844, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2282, 32} \[ -\frac{1}{e^x+1} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 32
Rubi steps
\begin{align*} \int \frac{e^x}{1+2 e^x+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,e^x\right )\\ &=-\frac{1}{1+e^x}\\ \end{align*}
Mathematica [A] time = 0.0074786, size = 9, normalized size = 1. \[ -\frac{1}{e^x+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 9, normalized size = 1. \begin{align*} - \left ( 1+{{\rm e}^{x}} \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971287, size = 11, normalized size = 1.22 \begin{align*} -\frac{1}{e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.806071, size = 19, normalized size = 2.11 \begin{align*} -\frac{1}{e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.079009, size = 7, normalized size = 0.78 \begin{align*} - \frac{1}{e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24297, size = 11, normalized size = 1.22 \begin{align*} -\frac{1}{e^{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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