3.522 \(\int \frac{1}{b+a e^{n x}} \, dx\)

Optimal. Leaf size=24 \[ \frac{x}{b}-\frac{\log \left (a e^{n x}+b\right )}{b n} \]

[Out]

x/b - Log[b + a*E^(n*x)]/(b*n)

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Rubi [A]  time = 0.032037, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{x}{b}-\frac{\log \left (a e^{n x}+b\right )}{b n} \]

Antiderivative was successfully verified.

[In]  Int[(b + a*E^(n*x))^(-1),x]

[Out]

x/b - Log[b + a*E^(n*x)]/(b*n)

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Rubi in Sympy [A]  time = 3.13836, size = 22, normalized size = 0.92 \[ - \frac{\log{\left (a e^{n x} + b \right )}}{b n} + \frac{\log{\left (e^{n x} \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b+a*exp(n*x)),x)

[Out]

-log(a*exp(n*x) + b)/(b*n) + log(exp(n*x))/(b*n)

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Mathematica [A]  time = 0.00754968, size = 24, normalized size = 1. \[ \frac{x}{b}-\frac{\log \left (a e^{n x}+b\right )}{b n} \]

Antiderivative was successfully verified.

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

[Out]

x/b - Log[b + a*E^(n*x)]/(b*n)

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Maple [A]  time = 0.013, size = 31, normalized size = 1.3 \[{\frac{\ln \left ({{\rm e}^{nx}} \right ) }{nb}}-{\frac{\ln \left ( b+a{{\rm e}^{nx}} \right ) }{nb}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b+a*exp(n*x)),x)

[Out]

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

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Maxima [A]  time = 1.33471, size = 31, normalized size = 1.29 \[ \frac{x}{b} - \frac{\log \left (a e^{\left (n x\right )} + b\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*e^(n*x) + b),x, algorithm="maxima")

[Out]

x/b - log(a*e^(n*x) + b)/(b*n)

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Fricas [A]  time = 0.233617, size = 30, normalized size = 1.25 \[ \frac{n x - \log \left (a e^{\left (n x\right )} + b\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(n*x - log(a*e^(n*x) + b))/(b*n)

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Sympy [A]  time = 0.131794, size = 15, normalized size = 0.62 \[ \frac{x}{b} - \frac{\log{\left (e^{n x} + \frac{b}{a} \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b+a*exp(n*x)),x)

[Out]

x/b - log(exp(n*x) + b/a)/(b*n)

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GIAC/XCAS [A]  time = 0.200428, size = 32, normalized size = 1.33 \[ \frac{x}{b} - \frac{{\rm ln}\left ({\left | a e^{\left (n x\right )} + b \right |}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/b - ln(abs(a*e^(n*x) + b))/(b*n)