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]

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

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\log \left (a+b e^{c+d x}\right )}{b d} \]

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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fricas [A]  time = 0.40, size = 18, normalized size = 0.95 \[ \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \]

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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giac [A]  time = 0.34, size = 19, normalized size = 1.00 \[ \frac {\log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{b d} \]

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)

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maple [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {\ln \left (b \,{\mathrm e}^{d x +c}+a \right )}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 18, normalized size = 0.95 \[ \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \]

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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mupad [B]  time = 0.05, size = 18, normalized size = 0.95 \[ \frac {\ln \left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}{b\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*exp(c + d*x))/(b*d)

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sympy [A]  time = 0.13, size = 14, normalized size = 0.74 \[ \frac {\log {\left (\frac {a}{b} + e^{c + d x} \right )}}{b d} \]

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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