Optimal. Leaf size=27 \[ -\frac{x}{9}+\frac{e^{-x}}{3}+\frac{1}{9} \log \left (3-e^x\right ) \]
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Rubi [A] time = 0.0188416, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 44} \[ -\frac{x}{9}+\frac{e^{-x}}{3}+\frac{1}{9} \log \left (3-e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{-3 e^x+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{(-3+x) x^2} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{9 (-3+x)}-\frac{1}{3 x^2}-\frac{1}{9 x}\right ) \, dx,x,e^x\right )\\ &=\frac{e^{-x}}{3}-\frac{x}{9}+\frac{1}{9} \log \left (3-e^x\right )\\ \end{align*}
Mathematica [A] time = 0.018038, size = 23, normalized size = 0.85 \[ \frac{1}{9} \left (-x+3 e^{-x}+\log \left (3-e^x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 20, normalized size = 0.7 \begin{align*}{\frac{\ln \left ({{\rm e}^{x}}-3 \right ) }{9}}+{\frac{1}{3\,{{\rm e}^{x}}}}-{\frac{\ln \left ({{\rm e}^{x}} \right ) }{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975396, size = 23, normalized size = 0.85 \begin{align*} -\frac{1}{9} \, x + \frac{1}{3} \, e^{\left (-x\right )} + \frac{1}{9} \, \log \left (e^{x} - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.773875, size = 59, normalized size = 2.19 \begin{align*} -\frac{1}{9} \,{\left (x e^{x} - e^{x} \log \left (e^{x} - 3\right ) - 3\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.099519, size = 17, normalized size = 0.63 \begin{align*} - \frac{x}{9} + \frac{\log{\left (e^{x} - 3 \right )}}{9} + \frac{e^{- x}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24845, size = 24, normalized size = 0.89 \begin{align*} -\frac{1}{9} \, x + \frac{1}{3} \, e^{\left (-x\right )} + \frac{1}{9} \, \log \left ({\left | e^{x} - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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