Optimal. Leaf size=26 \[ \frac{x}{a}-\frac{\log \left (a+b e^{c+d x}\right )}{a d} \]
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Rubi [A] time = 0.0175739, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2282, 36, 29, 31} \[ \frac{x}{a}-\frac{\log \left (a+b e^{c+d x}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+b e^{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^{c+d x}\right )}{a d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^{c+d x}\right )}{a d}\\ &=\frac{x}{a}-\frac{\log \left (a+b e^{c+d x}\right )}{a d}\\ \end{align*}
Mathematica [A] time = 0.0039174, size = 26, normalized size = 1. \[ \frac{x}{a}-\frac{\log \left (a+b e^{c+d x}\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 35, normalized size = 1.4 \begin{align*}{\frac{\ln \left ({{\rm e}^{dx+c}} \right ) }{ad}}-{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04686, size = 43, normalized size = 1.65 \begin{align*} \frac{d x + c}{a d} - \frac{\log \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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Fricas [A] time = 1.49047, size = 51, normalized size = 1.96 \begin{align*} \frac{d x - \log \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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Sympy [A] time = 0.188579, size = 17, normalized size = 0.65 \begin{align*} \frac{x}{a} - \frac{\log{\left (\frac{a}{b} + e^{c + d x} \right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24266, size = 45, normalized size = 1.73 \begin{align*} \frac{d x + c}{a d} - \frac{\log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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