Optimal. Leaf size=61 \[ -\frac{b}{a^2 n \left (a+b e^{n x}\right )}+\frac{2 b \log \left (a+b e^{n x}\right )}{a^3 n}-\frac{2 b x}{a^3}-\frac{e^{-n x}}{a^2 n} \]
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Rubi [A] time = 0.0572723, antiderivative size = 61, 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} \[ -\frac{b}{a^2 n \left (a+b e^{n x}\right )}+\frac{2 b \log \left (a+b e^{n x}\right )}{a^3 n}-\frac{2 b x}{a^3}-\frac{e^{-n x}}{a^2 n} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{-n x}}{\left (a+b e^{n x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac{e^{-n x}}{a^2 n}-\frac{b}{a^2 \left (a+b e^{n x}\right ) n}-\frac{2 b x}{a^3}+\frac{2 b \log \left (a+b e^{n x}\right )}{a^3 n}\\ \end{align*}
Mathematica [A] time = 0.107794, size = 49, normalized size = 0.8 \[ -\frac{a \left (\frac{b}{a+b e^{n x}}+e^{-n x}\right )-2 b \log \left (a+b e^{n x}\right )+2 b n x}{a^3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 67, normalized size = 1.1 \begin{align*} -{\frac{1}{{a}^{2}{{\rm e}^{nx}}n}}-2\,{\frac{b\ln \left ({{\rm e}^{nx}} \right ) }{{a}^{3}n}}-{\frac{b}{{a}^{2} \left ( a+b{{\rm e}^{nx}} \right ) n}}+2\,{\frac{b\ln \left ( a+b{{\rm e}^{nx}} \right ) }{{a}^{3}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10071, size = 77, normalized size = 1.26 \begin{align*} \frac{b^{2}}{{\left (a^{4} e^{\left (-n x\right )} + a^{3} b\right )} n} - \frac{e^{\left (-n x\right )}}{a^{2} n} + \frac{2 \, b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52061, size = 198, normalized size = 3.25 \begin{align*} -\frac{2 \, b^{2} n x e^{\left (2 \, n x\right )} + a^{2} + 2 \,{\left (a b n x + a b\right )} e^{\left (n x\right )} - 2 \,{\left (b^{2} e^{\left (2 \, n x\right )} + a b e^{\left (n x\right )}\right )} \log \left (b e^{\left (n x\right )} + a\right )}{a^{3} b n e^{\left (2 \, n x\right )} + a^{4} n e^{\left (n x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.201925, size = 78, normalized size = 1.28 \begin{align*} - \frac{b}{a^{3} n + a^{2} b n e^{n x}} + \begin{cases} - \frac{e^{- n x}}{a^{2} n} & \text{for}\: a^{2} n \neq 0 \\x \left (\frac{2 b}{a^{3}} + \frac{a - 2 b}{a^{3}}\right ) & \text{otherwise} \end{cases} - \frac{2 b x}{a^{3}} + \frac{2 b \log{\left (\frac{a}{b} + e^{n x} \right )}}{a^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28641, size = 80, normalized size = 1.31 \begin{align*} -\frac{\frac{2 \, b n x}{a^{3}} - \frac{2 \, b \log \left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{3}} + \frac{2 \, b e^{\left (n x\right )} + a}{{\left (b e^{\left (2 \, n x\right )} + a e^{\left (n x\right )}\right )} a^{2}}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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