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 = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 44
Rule 2282
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.40, size = 21, normalized size = 0.78 \[ -\frac {1}{9} \, {\left (x e^{x} - e^{x} \log \left (e^{x} - 3\right ) - 3\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 18, normalized size = 0.67 \[ -\frac {1}{9} \, x + \frac {1}{3} \, e^{\left (-x\right )} + \frac {1}{9} \, \log \left ({\left | e^{x} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.74 \[ \frac {{\mathrm e}^{-x}}{3}+\frac {\ln \left ({\mathrm e}^{x}-3\right )}{9}-\frac {\ln \left ({\mathrm e}^{x}\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 17, normalized size = 0.63 \[ -\frac {1}{9} \, x + \frac {1}{3} \, e^{\left (-x\right )} + \frac {1}{9} \, \log \left (e^{x} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 17, normalized size = 0.63 \[ \frac {{\mathrm {e}}^{-x}}{3}-\frac {x}{9}+\frac {\ln \left ({\mathrm {e}}^x-3\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 17, normalized size = 0.63 \[ - \frac {x}{9} + \frac {\log {\left (e^{x} - 3 \right )}}{9} + \frac {e^{- x}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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