Optimal. Leaf size=40 \[ \frac{b \log \left (a+b e^{n x}\right )}{a^2 n}-\frac{b x}{a^2}-\frac{e^{-n x}}{a n} \]
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Rubi [A] time = 0.0438791, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2248, 44} \[ \frac{b \log \left (a+b e^{n x}\right )}{a^2 n}-\frac{b x}{a^2}-\frac{e^{-n x}}{a n} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{-n x}}{a+b e^{n x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac{e^{-n x}}{a n}-\frac{b x}{a^2}+\frac{b \log \left (a+b e^{n x}\right )}{a^2 n}\\ \end{align*}
Mathematica [A] time = 0.0382528, size = 34, normalized size = 0.85 \[ -\frac{-b \log \left (a+b e^{n x}\right )+a e^{-n x}+b n x}{a^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 47, normalized size = 1.2 \begin{align*} -{\frac{1}{a{{\rm e}^{nx}}n}}-{\frac{b\ln \left ({{\rm e}^{nx}} \right ) }{{a}^{2}n}}+{\frac{b\ln \left ( a+b{{\rm e}^{nx}} \right ) }{{a}^{2}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04324, size = 43, normalized size = 1.08 \begin{align*} -\frac{e^{\left (-n x\right )}}{a n} + \frac{b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53373, size = 95, normalized size = 2.38 \begin{align*} -\frac{{\left (b n x e^{\left (n x\right )} - b e^{\left (n x\right )} \log \left (b e^{\left (n x\right )} + a\right ) + a\right )} e^{\left (-n x\right )}}{a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.361883, size = 49, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{e^{- n x}}{a n} & \text{for}\: a n \neq 0 \\x \left (\frac{b}{a^{2}} + \frac{a - b}{a^{2}}\right ) & \text{otherwise} \end{cases} - \frac{b x}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + e^{n x} \right )}}{a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26291, size = 51, normalized size = 1.27 \begin{align*} -\frac{\frac{b n x}{a^{2}} + \frac{e^{\left (-n x\right )}}{a} - \frac{b \log \left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{2}}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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