Optimal. Leaf size=9 \[ -\frac {1}{e^x+1} \]
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Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 32
Rule 2282
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*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ -\frac {1}{e^x+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 8, normalized size = 0.89 \[ -\frac {1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 8, normalized size = 0.89 \[ -\frac {1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 9, normalized size = 1.00 \[ -\frac {1}{{\mathrm e}^{x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 8, normalized size = 0.89 \[ -\frac {1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 8, normalized size = 0.89 \[ -\frac {1}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 7, normalized size = 0.78 \[ - \frac {1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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