Optimal. Leaf size=24 \[ \frac{x}{b}-\frac{\log \left (a e^{n x}+b\right )}{b n} \]
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Rubi [A] time = 0.0161057, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2282, 36, 29, 31} \[ \frac{x}{b}-\frac{\log \left (a e^{n x}+b\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{b+a e^{n x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (b+a x)} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^{n x}\right )}{b n}-\frac{a \operatorname{Subst}\left (\int \frac{1}{b+a x} \, dx,x,e^{n x}\right )}{b n}\\ &=\frac{x}{b}-\frac{\log \left (b+a e^{n x}\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0055943, size = 24, normalized size = 1. \[ \frac{x}{b}-\frac{\log \left (a e^{n x}+b\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 31, normalized size = 1.3 \begin{align*}{\frac{\ln \left ({{\rm e}^{nx}} \right ) }{nb}}-{\frac{\ln \left ( b+a{{\rm e}^{nx}} \right ) }{nb}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935134, size = 31, normalized size = 1.29 \begin{align*} \frac{x}{b} - \frac{\log \left (a e^{\left (n x\right )} + b\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84173, size = 46, normalized size = 1.92 \begin{align*} \frac{n x - \log \left (a e^{\left (n x\right )} + b\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112088, size = 15, normalized size = 0.62 \begin{align*} \frac{x}{b} - \frac{\log{\left (e^{n x} + \frac{b}{a} \right )}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09529, size = 32, normalized size = 1.33 \begin{align*} \frac{x}{b} - \frac{\log \left ({\left | a e^{\left (n x\right )} + b \right |}\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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