Optimal. Leaf size=78 \[ -\frac{1}{a^2 d \left (a+b e^{-c-d x}\right )}+\frac{\log \left (a+b e^{-c-d x}\right )}{a^3 d}+\frac{x}{a^3}-\frac{1}{2 a d \left (a+b e^{-c-d x}\right )^2} \]
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Rubi [A] time = 0.0467423, antiderivative size = 78, 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} \[ -\frac{1}{a^2 d \left (a+b e^{-c-d x}\right )}+\frac{\log \left (a+b e^{-c-d x}\right )}{a^3 d}+\frac{x}{a^3}-\frac{1}{2 a d \left (a+b e^{-c-d x}\right )^2} \]
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 )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^3} \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{b}{a (a+b x)^3}-\frac{b}{a^2 (a+b x)^2}-\frac{b}{a^3 (a+b x)}\right ) \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{1}{2 a d \left (a+b e^{-c-d x}\right )^2}-\frac{1}{a^2 d \left (a+b e^{-c-d x}\right )}+\frac{x}{a^3}+\frac{\log \left (a+b e^{-c-d x}\right )}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.0905639, size = 54, normalized size = 0.69 \[ \frac{\frac{b \left (4 a e^{c+d x}+3 b\right )}{\left (a e^{c+d x}+b\right )^2}+2 \log \left (a e^{c+d x}+b\right )}{2 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 87, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ({{\rm e}^{-dx-c}} \right ) }{d{a}^{3}}}+{\frac{\ln \left ( a+b{{\rm e}^{-dx-c}} \right ) }{d{a}^{3}}}-{\frac{1}{{a}^{2}d \left ( a+b{{\rm e}^{-dx-c}} \right ) }}-{\frac{1}{2\,ad \left ( a+b{{\rm e}^{-dx-c}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06493, size = 124, normalized size = 1.59 \begin{align*} -\frac{2 \, b e^{\left (-d x - c\right )} + 3 \, a}{2 \,{\left (2 \, a^{3} b e^{\left (-d x - c\right )} + a^{2} b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{4}\right )} d} + \frac{d x + c}{a^{3} d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5306, size = 309, normalized size = 3.96 \begin{align*} \frac{2 \, b^{2} d x e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, a^{2} d x - 3 \, a^{2} + 2 \,{\left (2 \, a b d x - a b\right )} e^{\left (-d x - c\right )} + 2 \,{\left (2 \, a b e^{\left (-d x - c\right )} + b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{2}\right )} \log \left (b e^{\left (-d x - c\right )} + a\right )}{2 \,{\left (2 \, a^{4} b d e^{\left (-d x - c\right )} + a^{3} b^{2} d e^{\left (-2 \, d x - 2 \, c\right )} + a^{5} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.364033, size = 85, normalized size = 1.09 \begin{align*} \frac{- 3 a - 2 b e^{- c - d x}}{2 a^{4} d + 4 a^{3} b d e^{- c - d x} + 2 a^{2} b^{2} d e^{- 2 c - 2 d x}} + \frac{x}{a^{3}} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35954, size = 104, normalized size = 1.33 \begin{align*} \frac{d x + c}{a^{3} d} + \frac{\log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a^{3} d} - \frac{2 \, a b e^{\left (-d x - c\right )} + 3 \, a^{2}}{2 \,{\left (b e^{\left (-d x - c\right )} + a\right )}^{2} a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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