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

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 = 0.0699595, 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 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a^2 d}+\frac{x}{a^2}-\frac{1}{a d \left (a+b e^{-c-d x}\right )} \]

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)

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Rubi in Sympy [A]  time = 13.867, size = 94, normalized size = 1.81 \[ - \frac{e^{- c - d x} e^{c + d x}}{a d \left (a + b e^{- c - d x}\right )} + \frac{e^{- c - d x} e^{c + d x} \log{\left (a + b e^{- c - d x} \right )}}{a^{2} d} - \frac{e^{- c - d x} e^{c + d x} \log{\left (e^{- c - d x} \right )}}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*exp(-d*x-c))**2,x)

[Out]

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

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Mathematica [A]  time = 0.0511566, 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 = 0.003, size = 64, normalized size = 1.2 \[ -{\frac{\ln \left ({{\rm e}^{-dx-c}} \right ) }{d{a}^{2}}}+{\frac{\ln \left ( a+b{{\rm e}^{-dx-c}} \right ) }{d{a}^{2}}}-{\frac{1}{ad \left ( a+b{{\rm e}^{-dx-c}} \right ) }} \]

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 = 0.861881, size = 77, normalized size = 1.48 \[ -\frac{1}{{\left (a b e^{\left (-d x - c\right )} + a^{2}\right )} d} + \frac{d x + c}{a^{2} d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(-d*x - c) + a)^(-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 = 0.259679, size = 99, normalized size = 1.9 \[ \frac{b d x e^{\left (-d x - c\right )} + a d x +{\left (b e^{\left (-d x - c\right )} + a\right )} \log \left (b e^{\left (-d x - c\right )} + a\right ) - a}{a^{2} b d e^{\left (-d x - c\right )} + a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(-d*x - c) + a)^(-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 = 0.334785, size = 42, normalized size = 0.81 \[ - \frac{1}{a^{2} d + a b d e^{- c - d x}} + \frac{x}{a^{2}} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a^{2} d} \]

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/XCAS [A]  time = 0.256108, size = 78, normalized size = 1.5 \[ \frac{d x + c}{a^{2} d} + \frac{{\rm ln}\left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a^{2} d} - \frac{1}{{\left (b e^{\left (-d x - c\right )} + a\right )} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(-d*x - c) + a)^(-2),x, algorithm="giac")

[Out]

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