3.1.4 \(\int \frac {1}{a+b e^{c+d x}} \, dx\) [4]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2320, 36, 29, 31} \begin {gather*} \frac {x}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a d} \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*}

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Mathematica [A]
time = 0.02, size = 42, normalized size = 1.62 \begin {gather*} \frac {\log \left (e^{c+d x}\right )}{a d}-\frac {\log \left (a^2 d+a b d 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[E^(c + d*x)]/(a*d) - Log[a^2*d + a*b*d*E^(c + d*x)]/(a*d)

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Maple [A]
time = 0.02, size = 33, normalized size = 1.27

method	result	size
norman	\(\frac {x}{a}-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a d}\)	\(26\)
derivativedivides	\(\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a}}{d}\)	\(33\)
default	\(\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a}}{d}\)	\(33\)
risch	\(\frac {x}{a}+\frac {c}{a d}-\frac {\ln \left ({\mathrm e}^{d x +c}+\frac {a}{b}\right )}{a d}\)	\(36\)

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)))

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Maxima [A]
time = 0.30, size = 32, normalized size = 1.23 \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)

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Fricas [A]
time = 0.42, size = 24, normalized size = 0.92 \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)

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Sympy [A]
time = 0.05, size = 17, normalized size = 0.65 \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)

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Giac [A]
time = 1.78, size = 31, normalized size = 1.19 \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

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Mupad [B]
time = 3.50, size = 24, normalized size = 0.92 \begin {gather*} -\frac {\ln \left (a+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\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(d*x)*exp(c)) - d*x)/(a*d)

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