Optimal. Leaf size=26 \[ \frac{\log \left (a+b e^{c-d x}\right )}{a d}+\frac{x}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0208319, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2282, 36, 29, 31} \[ \frac{\log \left (a+b e^{c-d x}\right )}{a d}+\frac{x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
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 = 0.0153459, size = 21, normalized size = 0.81 \[ \frac{\log \left (a e^{d x}+b e^c\right )}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 37, 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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08387, size = 46, normalized size = 1.77 \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.50451, size = 53, normalized size = 2.04 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.138286, 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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.33411, size = 47, normalized size = 1.81 \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.
[In]
[Out]