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

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

[Out]

x/a + Log[a + b*E^(-c - d*x)]/(a*d)

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Rubi [A]  time = 0.0439113, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a d}+\frac{x}{a} \]

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)

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Rubi in Sympy [A]  time = 8.75816, size = 60, normalized size = 2.14 \[ \frac{e^{- c - d x} e^{c + d x} \log{\left (a + b e^{- c - d x} \right )}}{a d} - \frac{e^{- c - d x} e^{c + d x} \log{\left (e^{- c - d x} \right )}}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0130159, size = 19, normalized size = 0.68 \[ \frac{\log \left (a e^{c+d x}+b\right )}{a d} \]

Antiderivative was successfully verified.

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

[Out]

Log[b + a*E^(c + d*x)]/(a*d)

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Maple [A]  time = 0.004, size = 41, normalized size = 1.5 \[ -{\frac{\ln \left ({{\rm e}^{-dx-c}} \right ) }{da}}+{\frac{\ln \left ( a+b{{\rm e}^{-dx-c}} \right ) }{da}} \]

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 = 0.756931, size = 46, normalized size = 1.64 \[ \frac{d x + c}{a d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*e^(-d*x - c) + a),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 = 0.258171, size = 34, normalized size = 1.21 \[ \frac{d x + \log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*e^(-d*x - c) + a),x, algorithm="fricas")

[Out]

(d*x + log(b*e^(-d*x - c) + a))/(a*d)

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Sympy [A]  time = 0.254066, size = 19, normalized size = 0.68 \[ \frac{x}{a} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a d} \]

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/XCAS [A]  time = 0.285751, size = 47, normalized size = 1.68 \[ \frac{d x + c}{a d} + \frac{{\rm ln}\left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*e^(-d*x - c) + a),x, algorithm="giac")

[Out]

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