3.7 \(\int \frac{1}{a+b e^{c-d x}} \, dx\)

Optimal. Leaf size=26 \[ \frac{\log \left (a+b e^{c-d x}\right )}{a d}+\frac{x}{a} \]

[Out]

x/a + Log[a + b*E^(c - d*x)]/(a*d)

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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]

Int[(a + b*E^(c - d*x))^(-1),x]

[Out]

x/a + Log[a + b*E^(c - d*x)]/(a*d)

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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{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]

Integrate[(a + b*E^(c - d*x))^(-1),x]

[Out]

Log[b*E^c + a*E^(d*x)]/(a*d)

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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]

int(1/(a+b*exp(-d*x+c)),x)

[Out]

-1/d/a*ln(exp(-d*x+c))+ln(a+b*exp(-d*x+c))/a/d

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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]

integrate(1/(a+b*exp(-d*x+c)),x, algorithm="maxima")

[Out]

(d*x - c)/(a*d) + log(b*e^(-d*x + c) + a)/(a*d)

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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]

integrate(1/(a+b*exp(-d*x+c)),x, algorithm="fricas")

[Out]

(d*x + log(b*e^(-d*x + c) + a))/(a*d)

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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]

integrate(1/(a+b*exp(-d*x+c)),x)

[Out]

x/a + log(a/b + exp(c - d*x))/(a*d)

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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]

integrate(1/(a+b*exp(-d*x+c)),x, algorithm="giac")

[Out]

(d*x - c)/(a*d) + log(abs(b*e^(-d*x + c) + a))/(a*d)