Optimal. Leaf size=83 \[ -\frac{2 b}{a^3 n \left (a+b e^{n x}\right )}-\frac{b}{2 a^2 n \left (a+b e^{n x}\right )^2}+\frac{3 b \log \left (a+b e^{n x}\right )}{a^4 n}-\frac{3 b x}{a^4}-\frac{e^{-n x}}{a^3 n} \]
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Rubi [A] time = 0.0683861, antiderivative size = 83, 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} \[ -\frac{2 b}{a^3 n \left (a+b e^{n x}\right )}-\frac{b}{2 a^2 n \left (a+b e^{n x}\right )^2}+\frac{3 b \log \left (a+b e^{n x}\right )}{a^4 n}-\frac{3 b x}{a^4}-\frac{e^{-n x}}{a^3 n} \]
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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^3} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^2}-\frac{3 b}{a^4 x}+\frac{b^2}{a^2 (a+b x)^3}+\frac{2 b^2}{a^3 (a+b x)^2}+\frac{3 b^2}{a^4 (a+b x)}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac{e^{-n x}}{a^3 n}-\frac{b}{2 a^2 \left (a+b e^{n x}\right )^2 n}-\frac{2 b}{a^3 \left (a+b e^{n x}\right ) n}-\frac{3 b x}{a^4}+\frac{3 b \log \left (a+b e^{n x}\right )}{a^4 n}\\ \end{align*}
Mathematica [A] time = 0.133166, size = 69, normalized size = 0.83 \[ -\frac{\frac{a^2 b}{\left (a+b e^{n x}\right )^2}+\frac{4 a b}{a+b e^{n x}}-6 b \log \left (a+b e^{n x}\right )+2 a e^{-n x}+6 b n x}{2 a^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 86, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{3}{{\rm e}^{nx}}n}}-3\,{\frac{b\ln \left ({{\rm e}^{nx}} \right ) }{n{a}^{4}}}-{\frac{b}{2\,{a}^{2} \left ( a+b{{\rm e}^{nx}} \right ) ^{2}n}}+3\,{\frac{b\ln \left ( a+b{{\rm e}^{nx}} \right ) }{n{a}^{4}}}-2\,{\frac{b}{{a}^{3} \left ( a+b{{\rm e}^{nx}} \right ) n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15054, size = 115, normalized size = 1.39 \begin{align*} \frac{6 \, a b^{2} e^{\left (-n x\right )} + 5 \, b^{3}}{2 \,{\left (2 \, a^{5} b e^{\left (-n x\right )} + a^{6} e^{\left (-2 \, n x\right )} + a^{4} b^{2}\right )} n} - \frac{e^{\left (-n x\right )}}{a^{3} n} + \frac{3 \, b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59791, size = 328, normalized size = 3.95 \begin{align*} -\frac{6 \, b^{3} n x e^{\left (3 \, n x\right )} + 2 \, a^{3} + 6 \,{\left (2 \, a b^{2} n x + a b^{2}\right )} e^{\left (2 \, n x\right )} + 3 \,{\left (2 \, a^{2} b n x + 3 \, a^{2} b\right )} e^{\left (n x\right )} - 6 \,{\left (b^{3} e^{\left (3 \, n x\right )} + 2 \, a b^{2} e^{\left (2 \, n x\right )} + a^{2} b e^{\left (n x\right )}\right )} \log \left (b e^{\left (n x\right )} + a\right )}{2 \,{\left (a^{4} b^{2} n e^{\left (3 \, n x\right )} + 2 \, a^{5} b n e^{\left (2 \, n x\right )} + a^{6} n e^{\left (n x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.235815, size = 114, normalized size = 1.37 \begin{align*} \frac{- 5 a b - 4 b^{2} e^{n x}}{2 a^{5} n + 4 a^{4} b n e^{n x} + 2 a^{3} b^{2} n e^{2 n x}} + \begin{cases} - \frac{e^{- n x}}{a^{3} n} & \text{for}\: a^{3} n \neq 0 \\x \left (\frac{3 b}{a^{4}} + \frac{a - 3 b}{a^{4}}\right ) & \text{otherwise} \end{cases} - \frac{3 b x}{a^{4}} + \frac{3 b \log{\left (\frac{a}{b} + e^{n x} \right )}}{a^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27695, size = 103, normalized size = 1.24 \begin{align*} -\frac{\frac{6 \, b n x}{a^{4}} - \frac{6 \, b \log \left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{4}} + \frac{{\left (6 \, a b^{2} e^{\left (2 \, n x\right )} + 9 \, a^{2} b e^{\left (n x\right )} + 2 \, a^{3}\right )} e^{\left (-n x\right )}}{{\left (b e^{\left (n x\right )} + a\right )}^{2} a^{4}}}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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