3.16 \(\int \frac{1}{(a+b e^{-c-d x})^2} \, dx\)

Optimal. Leaf size=52 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a^2 d}+\frac{x}{a^2}-\frac{1}{a d \left (a+b e^{-c-d x}\right )} \]

[Out]

-(1/(a*d*(a + b*E^(-c - d*x)))) + x/a^2 + Log[a + b*E^(-c - d*x)]/(a^2*d)

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Rubi [A]  time = 0.0347173, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2282, 44} \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a^2 d}+\frac{x}{a^2}-\frac{1}{a d \left (a+b e^{-c-d x}\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^(-c - d*x))^(-2),x]

[Out]

-(1/(a*d*(a + b*E^(-c - d*x)))) + x/a^2 + Log[a + b*E^(-c - d*x)]/(a^2*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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b e^{-c-d x}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{1}{a d \left (a+b e^{-c-d x}\right )}+\frac{x}{a^2}+\frac{\log \left (a+b e^{-c-d x}\right )}{a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0565474, size = 35, normalized size = 0.67 \[ \frac{\frac{b}{a e^{c+d x}+b}+\log \left (a e^{c+d x}+b\right )}{a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^(-c - d*x))^(-2),x]

[Out]

(b/(b + a*E^(c + d*x)) + Log[b + a*E^(c + d*x)])/(a^2*d)

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Maple [A]  time = 0.001, size = 64, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ({{\rm e}^{-dx-c}} \right ) }{{a}^{2}d}}+{\frac{\ln \left ( a+b{{\rm e}^{-dx-c}} \right ) }{{a}^{2}d}}-{\frac{1}{ad \left ( a+b{{\rm e}^{-dx-c}} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*exp(-d*x-c))^2,x)

[Out]

-1/d/a^2*ln(exp(-d*x-c))+ln(a+b*exp(-d*x-c))/a^2/d-1/a/d/(a+b*exp(-d*x-c))

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Maxima [A]  time = 1.05075, size = 77, normalized size = 1.48 \begin{align*} -\frac{1}{{\left (a b e^{\left (-d x - c\right )} + a^{2}\right )} d} + \frac{d x + c}{a^{2} d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c))^2,x, algorithm="maxima")

[Out]

-1/((a*b*e^(-d*x - c) + a^2)*d) + (d*x + c)/(a^2*d) + log(b*e^(-d*x - c) + a)/(a^2*d)

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Fricas [A]  time = 1.48949, size = 151, normalized size = 2.9 \begin{align*} \frac{b d x e^{\left (-d x - c\right )} + a d x +{\left (b e^{\left (-d x - c\right )} + a\right )} \log \left (b e^{\left (-d x - c\right )} + a\right ) - a}{a^{2} b d e^{\left (-d x - c\right )} + a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c))^2,x, algorithm="fricas")

[Out]

(b*d*x*e^(-d*x - c) + a*d*x + (b*e^(-d*x - c) + a)*log(b*e^(-d*x - c) + a) - a)/(a^2*b*d*e^(-d*x - c) + a^3*d)

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Sympy [A]  time = 0.274332, size = 42, normalized size = 0.81 \begin{align*} - \frac{1}{a^{2} d + a b d e^{- c - d x}} + \frac{x}{a^{2}} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c))**2,x)

[Out]

-1/(a**2*d + a*b*d*exp(-c - d*x)) + x/a**2 + log(a/b + exp(-c - d*x))/(a**2*d)

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Giac [A]  time = 1.26487, size = 78, normalized size = 1.5 \begin{align*} \frac{d x + c}{a^{2} d} + \frac{\log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a^{2} d} - \frac{1}{{\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))^2,x, algorithm="giac")

[Out]

(d*x + c)/(a^2*d) + log(abs(b*e^(-d*x - c) + a))/(a^2*d) - 1/((b*e^(-d*x - c) + a)*a*d)