3.2 \(\int \frac{e^x}{a+b e^x} \, dx\)

Optimal. Leaf size=12 \[ \frac{\log \left (a+b e^x\right )}{b} \]

[Out]

Log[a + b*E^x]/b

________________________________________________________________________________________

Rubi [A]  time = 0.0197353, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2246, 31} \[ \frac{\log \left (a+b e^x\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[E^x/(a + b*E^x),x]

[Out]

Log[a + b*E^x]/b

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{e^x}{a+b e^x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^x\right )\\ &=\frac{\log \left (a+b e^x\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0049704, size = 12, normalized size = 1. \[ \frac{\log \left (a+b e^x\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(a + b*E^x),x]

[Out]

Log[a + b*E^x]/b

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 12, normalized size = 1. \begin{align*}{\frac{\ln \left ( a+b{{\rm e}^{x}} \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(a+b*exp(x)),x)

[Out]

ln(a+b*exp(x))/b

________________________________________________________________________________________

Maxima [A]  time = 1.05088, size = 15, normalized size = 1.25 \begin{align*} \frac{\log \left (b e^{x} + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(a+b*exp(x)),x, algorithm="maxima")

[Out]

log(b*e^x + a)/b

________________________________________________________________________________________

Fricas [A]  time = 1.43886, size = 24, normalized size = 2. \begin{align*} \frac{\log \left (b e^{x} + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(a+b*exp(x)),x, algorithm="fricas")

[Out]

log(b*e^x + a)/b

________________________________________________________________________________________

Sympy [A]  time = 0.124032, size = 8, normalized size = 0.67 \begin{align*} \frac{\log{\left (\frac{a}{b} + e^{x} \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(a+b*exp(x)),x)

[Out]

log(a/b + exp(x))/b

________________________________________________________________________________________

Giac [A]  time = 1.20525, size = 16, normalized size = 1.33 \begin{align*} \frac{\log \left ({\left | b e^{x} + a \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(a+b*exp(x)),x, algorithm="giac")

[Out]

log(abs(b*e^x + a))/b