Optimal. Leaf size=24 \[ \frac{e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0692306, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2247, 2246, 31} \[ \frac{e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2247
Rule 2246
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{d x}}{a+b e^{c+d x}} \, dx &=e^{-c} \int \frac{e^{c+d x}}{a+b e^{c+d x}} \, dx\\ &=\frac{e^{-c} \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{e^{-c} \log \left (a+b e^{c+d x}\right )}{b d}\\ \end{align*}
Mathematica [A] time = 0.0117087, size = 24, normalized size = 1. \[ \frac{e^{-c} \log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 23, normalized size = 1. \begin{align*}{\frac{\ln \left ( a+b{{\rm e}^{dx}}{{\rm e}^{c}} \right ) }{bd{{\rm e}^{c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06388, size = 30, normalized size = 1.25 \begin{align*} \frac{e^{\left (-c\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46597, size = 50, normalized size = 2.08 \begin{align*} \frac{e^{\left (-c\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.177016, size = 19, normalized size = 0.79 \begin{align*} \frac{e^{- c} \log{\left (\frac{a e^{- c}}{b} + e^{d x} \right )}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2031, size = 31, normalized size = 1.29 \begin{align*} \frac{e^{\left (-c\right )} \log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]