Optimal. Leaf size=83 \[ \frac {3 b \log \left (a+b e^{n x}\right )}{a^4 n}-\frac {3 b x}{a^4}-\frac {2 b}{a^3 n \left (a+b e^{n x}\right )}-\frac {e^{-n x}}{a^3 n}-\frac {b}{2 a^2 n \left (a+b e^{n x}\right )^2} \]
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Rubi [A] time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 44
Rule 2248
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*}
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Mathematica [A] time = 0.14, 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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fricas [A] time = 0.43, size = 140, normalized size = 1.69 \[ -\frac {6 \, b^{3} n x e^{\left (3 \, n x\right )} + 2 \, a^{3} + 6 \, {\left (2 \, a b^{2} n x + a b^{2}\right )} e^{\left (2 \, n x\right )} + 3 \, {\left (2 \, a^{2} b n x + 3 \, a^{2} b\right )} e^{\left (n x\right )} - 6 \, {\left (b^{3} e^{\left (3 \, n x\right )} + 2 \, a b^{2} e^{\left (2 \, n x\right )} + a^{2} b e^{\left (n x\right )}\right )} \log \left (b e^{\left (n x\right )} + a\right )}{2 \, {\left (a^{4} b^{2} n e^{\left (3 \, n x\right )} + 2 \, a^{5} b n e^{\left (2 \, n x\right )} + a^{6} n e^{\left (n x\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 76, normalized size = 0.92 \[ -\frac {\frac {6 \, b n x}{a^{4}} - \frac {6 \, b \log \left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{4}} + \frac {{\left (6 \, a b^{2} e^{\left (2 \, n x\right )} + 9 \, a^{2} b e^{\left (n x\right )} + 2 \, a^{3}\right )} e^{\left (-n x\right )}}{{\left (b e^{\left (n x\right )} + a\right )}^{2} a^{4}}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 86, normalized size = 1.04 \[ -\frac {b}{2 \left (b \,{\mathrm e}^{n x}+a \right )^{2} a^{2} n}-\frac {2 b}{\left (b \,{\mathrm e}^{n x}+a \right ) a^{3} n}-\frac {{\mathrm e}^{-n x}}{a^{3} n}+\frac {3 b \ln \left (b \,{\mathrm e}^{n x}+a \right )}{a^{4} n}-\frac {3 b \ln \left ({\mathrm e}^{n x}\right )}{a^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 85, normalized size = 1.02 \[ \frac {6 \, a b^{2} e^{\left (-n x\right )} + 5 \, b^{3}}{2 \, {\left (2 \, a^{5} b e^{\left (-n x\right )} + a^{6} e^{\left (-2 \, n x\right )} + a^{4} b^{2}\right )} n} - \frac {e^{\left (-n x\right )}}{a^{3} n} + \frac {3 \, b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 104, normalized size = 1.25 \[ \frac {\frac {6\,b^2\,{\mathrm {e}}^{2\,n\,x}}{a^3\,n}-\frac {1}{a\,n}+\frac {9\,b^3\,{\mathrm {e}}^{3\,n\,x}}{2\,a^4\,n}}{{\mathrm {e}}^{n\,x}\,a^2+2\,{\mathrm {e}}^{2\,n\,x}\,a\,b+{\mathrm {e}}^{3\,n\,x}\,b^2}-\frac {3\,b\,\ln \left ({\mathrm {e}}^{n\,x}\right )}{a^4\,n}+\frac {3\,b\,\ln \left (a+b\,{\mathrm {e}}^{n\,x}\right )}{a^4\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 95, normalized size = 1.14 \[ \frac {6 a b^{2} e^{- n x} + 5 b^{3}}{2 a^{6} n e^{- 2 n x} + 4 a^{5} b n e^{- n x} + 2 a^{4} b^{2} n} + \begin {cases} - \frac {e^{- n x}}{a^{3} n} & \text {for}\: a^{3} n \neq 0 \\\frac {x}{a^{3}} & \text {otherwise} \end {cases} + \frac {3 b \log {\left (e^{- n x} + \frac {b}{a} \right )}}{a^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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