Optimal. Leaf size=13 \[ -\frac{1}{\left (3^x+1\right ) \log (3)} \]
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Rubi [A] time = 0.0120727, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 32} \[ -\frac{1}{\left (3^x+1\right ) \log (3)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{2+3^{-x}+3^x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1+x)^2} \, dx,x,3^x\right )}{\log (3)}\\ &=-\frac{1}{\left (1+3^x\right ) \log (3)}\\ \end{align*}
Mathematica [A] time = 0.010441, size = 13, normalized size = 1. \[ -\frac{1}{\left (3^x+1\right ) \log (3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{ \left ( 1+{3}^{x} \right ) \ln \left ( 3 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987349, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{{\left (\frac{1}{3^{x}} + 1\right )} \log \left (3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22765, size = 35, normalized size = 2.69 \begin{align*} -\frac{1}{3^{x} \log \left (3\right ) + \log \left (3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.088442, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{3^{x} \log{\left (3 \right )} + \log{\left (3 \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19945, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{{\left (3^{x} + 1\right )} \log \left (3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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