3.1.8 \(\int \frac {1}{a+b e^{-c-d x}} \, dx\) [8]

Optimal. Leaf size=28 \[ \frac {x}{a}+\frac {\log \left (a+b e^{-c-d x}\right )}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 36, 29, 31} \begin {gather*} \frac {\log \left (a+b e^{-c-d x}\right )}{a d}+\frac {x}{a} \end {gather*}

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 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]

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 2320

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]]]

Rubi steps

\begin {align*} \int \frac {1}{a+b e^{-c-d x}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{-c-d x}\right )}{a d}+\frac {b \text {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 = 0.02, size = 19, normalized size = 0.68 \begin {gather*} \frac {\log \left (b+a e^{c+d x}\right )}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^(-c - d*x))^(-1),x]

[Out]

Log[b + a*E^(c + d*x)]/(a*d)

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 40, normalized size = 1.43

method	result	size
norman	\(\frac {x}{a}+\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a d}\)	\(28\)
risch	\(\frac {x}{a}+\frac {c}{a d}+\frac {\ln \left ({\mathrm e}^{-d x -c}+\frac {a}{b}\right )}{a d}\)	\(38\)
derivativedivides	\(-\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a}+\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{a}}{d}\)	\(40\)
default	\(-\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a}+\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{a}}{d}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 34, normalized size = 1.21 \begin {gather*} \frac {d x + c}{a d} + \frac {\log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 25, normalized size = 0.89 \begin {gather*} \frac {d x + \log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.06, size = 19, normalized size = 0.68 \begin {gather*} \frac {x}{a} + \frac {\log {\left (\frac {a}{b} + e^{- c - d x} \right )}}{a d} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 2.25, size = 33, normalized size = 1.18 \begin {gather*} \frac {\frac {d x + c}{a} + \frac {\log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a}}{d} \end {gather*}

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 + log(abs(b*e^(-d*x - c) + a))/a)/d

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 25, normalized size = 0.89 \begin {gather*} \frac {\ln \left (a+b\,{\mathrm {e}}^{-c}\,{\mathrm {e}}^{-d\,x}\right )+d\,x}{a\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*exp(- c - d*x)),x)

[Out]

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

________________________________________________________________________________________