Optimal. Leaf size=27 \[ \frac{a}{b^2 \left (a+b e^x\right )}+\frac{\log \left (a+b e^x\right )}{b^2} \]
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Rubi [A] time = 0.0337294, antiderivative size = 27, 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} \[ \frac{a}{b^2 \left (a+b e^x\right )}+\frac{\log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{\left (a+b e^x\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{x}{(a+b x)^2} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx,x,e^x\right )\\ &=\frac{a}{b^2 \left (a+b e^x\right )}+\frac{\log \left (a+b e^x\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0240013, size = 24, normalized size = 0.89 \[ \frac{\frac{a}{a+b e^x}+\log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 26, normalized size = 1. \begin{align*}{\frac{a}{{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) }}+{\frac{\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 = 1.06359, size = 38, normalized size = 1.41 \begin{align*} \frac{a}{b^{3} e^{x} + a b^{2}} + \frac{\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 = 1.45637, size = 70, normalized size = 2.59 \begin{align*} \frac{{\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b^{3} e^{x} + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.164065, size = 24, normalized size = 0.89 \begin{align*} \frac{a}{a b^{2} + b^{3} e^{x}} + \frac{\log{\left (\frac{a}{b} + e^{x} \right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30259, size = 35, normalized size = 1.3 \begin{align*} \frac{\log \left ({\left | b e^{x} + a \right |}\right )}{b^{2}} + \frac{a}{{\left (b e^{x} + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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