Optimal. Leaf size=52 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a^2 d}+\frac{x}{a^2}-\frac{1}{a d \left (a+b e^{-c-d x}\right )} \]
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Rubi [A] time = 0.0347173, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2282, 44} \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a^2 d}+\frac{x}{a^2}-\frac{1}{a d \left (a+b e^{-c-d x}\right )} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b e^{-c-d x}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{1}{a d \left (a+b e^{-c-d x}\right )}+\frac{x}{a^2}+\frac{\log \left (a+b e^{-c-d x}\right )}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0565474, size = 35, normalized size = 0.67 \[ \frac{\frac{b}{a e^{c+d x}+b}+\log \left (a e^{c+d x}+b\right )}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 64, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ({{\rm e}^{-dx-c}} \right ) }{{a}^{2}d}}+{\frac{\ln \left ( a+b{{\rm e}^{-dx-c}} \right ) }{{a}^{2}d}}-{\frac{1}{ad \left ( a+b{{\rm e}^{-dx-c}} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05075, size = 77, normalized size = 1.48 \begin{align*} -\frac{1}{{\left (a b e^{\left (-d x - c\right )} + a^{2}\right )} d} + \frac{d x + c}{a^{2} d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48949, size = 151, normalized size = 2.9 \begin{align*} \frac{b d x e^{\left (-d x - c\right )} + a d x +{\left (b e^{\left (-d x - c\right )} + a\right )} \log \left (b e^{\left (-d x - c\right )} + a\right ) - a}{a^{2} b d e^{\left (-d x - c\right )} + a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.274332, size = 42, normalized size = 0.81 \begin{align*} - \frac{1}{a^{2} d + a b d e^{- c - d x}} + \frac{x}{a^{2}} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26487, size = 78, normalized size = 1.5 \begin{align*} \frac{d x + c}{a^{2} d} + \frac{\log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a^{2} d} - \frac{1}{{\left (b e^{\left (-d x - c\right )} + a\right )} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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