Optimal. Leaf size=22 \[ \frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2} \]
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Rubi [A] time = 0.0320372, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{a+b e^x} \, dx &=\operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,e^x\right )\\ &=\frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0152858, size = 22, normalized size = 1. \[ \frac{e^x}{b}-\frac{a \log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{x}}}{b}}-{\frac{a\ln \left ( a+b{{\rm e}^{x}} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19547, size = 27, normalized size = 1.23 \begin{align*} \frac{e^{x}}{b} - \frac{a \log \left (b e^{x} + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52141, size = 43, normalized size = 1.95 \begin{align*} \frac{b e^{x} - a \log \left (b e^{x} + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.134158, size = 20, normalized size = 0.91 \begin{align*} - \frac{a \log{\left (\frac{a}{b} + e^{x} \right )}}{b^{2}} + \begin{cases} \frac{e^{x}}{b} & \text{for}\: b \neq 0 \\\frac{x}{b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3437, size = 28, normalized size = 1.27 \begin{align*} \frac{e^{x}}{b} - \frac{a \log \left ({\left | b e^{x} + a \right |}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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