3.4 \(\int \frac{e^{c+d x}}{a+b e^{c+d x}} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log \left (a+b e^{c+d x}\right )}{b d} \]

[Out]

Log[a + b*E^(c + d*x)]/(b*d)

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Rubi [A]  time = 0.0359485, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2246, 31} \[ \frac{\log \left (a+b e^{c+d x}\right )}{b d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)/(a + b*E^(c + d*x)),x]

[Out]

Log[a + b*E^(c + d*x)]/(b*d)

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

Mathematica [A]  time = 0.0063149, size = 19, normalized size = 1. \[ \frac{\log \left (a+b e^{c+d x}\right )}{b d} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x)/(a + b*E^(c + d*x)),x]

[Out]

Log[a + b*E^(c + d*x)]/(b*d)

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Maple [A]  time = 0.003, size = 19, normalized size = 1. \begin{align*}{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10066, size = 24, normalized size = 1.26 \begin{align*} \frac{\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]

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

[Out]

log(b*e^(d*x + c) + a)/(b*d)

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Fricas [A]  time = 1.47187, size = 41, normalized size = 2.16 \begin{align*} \frac{\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]

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

[Out]

log(b*e^(d*x + c) + a)/(b*d)

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Sympy [A]  time = 0.218728, size = 14, normalized size = 0.74 \begin{align*} \frac{\log{\left (\frac{a}{b} + e^{c + d x} \right )}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a/b + exp(c + d*x))/(b*d)

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Giac [A]  time = 1.28083, size = 26, normalized size = 1.37 \begin{align*} \frac{\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]

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

[Out]

log(abs(b*e^(d*x + c) + a))/(b*d)