Optimal. Leaf size=24 \[ \frac{x}{a}-\frac{\log \left (a+b e^{m x}\right )}{a m} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0154089, 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} \[ \frac{x}{a}-\frac{\log \left (a+b e^{m x}\right )}{a m} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+b e^{m x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,e^{m x}\right )}{m}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^{m x}\right )}{a m}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^{m x}\right )}{a m}\\ &=\frac{x}{a}-\frac{\log \left (a+b e^{m x}\right )}{a m}\\ \end{align*}
Mathematica [A] time = 0.0054045, size = 24, normalized size = 1. \[ \frac{x}{a}-\frac{\log \left (a+b e^{m x}\right )}{a m} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 31, normalized size = 1.3 \begin{align*}{\frac{\ln \left ({{\rm e}^{mx}} \right ) }{ma}}-{\frac{\ln \left ( a+b{{\rm e}^{mx}} \right ) }{ma}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.928376, size = 31, normalized size = 1.29 \begin{align*} \frac{x}{a} - \frac{\log \left (b e^{\left (m x\right )} + a\right )}{a m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52804, size = 46, normalized size = 1.92 \begin{align*} \frac{m x - \log \left (b e^{\left (m x\right )} + a\right )}{a m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.168962, size = 15, normalized size = 0.62 \begin{align*} \frac{x}{a} - \frac{\log{\left (\frac{a}{b} + e^{m x} \right )}}{a m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07444, size = 32, normalized size = 1.33 \begin{align*} \frac{x}{a} - \frac{\log \left ({\left | b e^{\left (m x\right )} + a \right |}\right )}{a m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]