Optimal. Leaf size=40 \[ \frac {b \log \left (a+b e^{n x}\right )}{a^2 n}-\frac {b x}{a^2}-\frac {e^{-n x}}{a n} \]
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Rubi [A] time = 0.04, antiderivative size = 40, 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} \[ \frac {b \log \left (a+b e^{n x}\right )}{a^2 n}-\frac {b x}{a^2}-\frac {e^{-n x}}{a n} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{-n x}}{a+b e^{n x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac {e^{-n x}}{a n}-\frac {b x}{a^2}+\frac {b \log \left (a+b e^{n x}\right )}{a^2 n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 0.85 \[ -\frac {-b \log \left (a+b e^{n x}\right )+a e^{-n x}+b n x}{a^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 39, normalized size = 0.98 \[ -\frac {{\left (b n x e^{\left (n x\right )} - b e^{\left (n x\right )} \log \left (b e^{\left (n x\right )} + a\right ) + a\right )} e^{\left (-n x\right )}}{a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 38, normalized size = 0.95 \[ -\frac {\frac {b n x}{a^{2}} + \frac {e^{\left (-n x\right )}}{a} - \frac {b \log \left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{2}}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 1.18 \[ -\frac {{\mathrm e}^{-n x}}{a n}+\frac {b \ln \left (b \,{\mathrm e}^{n x}+a \right )}{a^{2} n}-\frac {b \ln \left ({\mathrm e}^{n x}\right )}{a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 32, normalized size = 0.80 \[ -\frac {e^{\left (-n x\right )}}{a n} + \frac {b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 38, normalized size = 0.95 \[ \frac {b\,\ln \left (a+b\,{\mathrm {e}}^{n\,x}\right )}{a^2\,n}-\frac {b\,x}{a^2}-\frac {{\mathrm {e}}^{-n\,x}}{a\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 34, normalized size = 0.85 \[ \begin {cases} - \frac {e^{- n x}}{a n} & \text {for}\: a n \neq 0 \\\frac {x}{a} & \text {otherwise} \end {cases} + \frac {b \log {\left (e^{- n x} + \frac {b}{a} \right )}}{a^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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