Optimal. Leaf size=18 \[ \frac{2}{3} \log \left (e^{2 x}+3\right )-\frac{x}{3} \]
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Rubi [A] time = 0.0258383, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2282, 72} \[ \frac{2}{3} \log \left (e^{2 x}+3\right )-\frac{x}{3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 72
Rubi steps
\begin{align*} \int \frac{-1+e^{2 x}}{3+e^{2 x}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+x}{x (3+x)} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{3 x}+\frac{4}{3 (3+x)}\right ) \, dx,x,e^{2 x}\right )\\ &=-\frac{x}{3}+\frac{2}{3} \log \left (3+e^{2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0086721, size = 18, normalized size = 1. \[ \frac{2}{3} \log \left (e^{2 x}+3\right )-\frac{x}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 18, normalized size = 1. \begin{align*} -{\frac{\ln \left ({{\rm e}^{2\,x}} \right ) }{6}}+{\frac{2\,\ln \left ( 3+{{\rm e}^{2\,x}} \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957969, size = 18, normalized size = 1. \begin{align*} -\frac{1}{3} \, x + \frac{2}{3} \, \log \left (e^{\left (2 \, x\right )} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.946902, size = 42, normalized size = 2.33 \begin{align*} -\frac{1}{3} \, x + \frac{2}{3} \, \log \left (e^{\left (2 \, x\right )} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087024, size = 14, normalized size = 0.78 \begin{align*} - \frac{x}{3} + \frac{2 \log{\left (e^{2 x} + 3 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2009, size = 18, normalized size = 1. \begin{align*} -\frac{1}{3} \, x + \frac{2}{3} \, \log \left (e^{\left (2 \, x\right )} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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