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

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

[Out]

x/a - Log[a + b*E^(m*x)]/(a*m)

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Rubi [A]  time = 0.0308992, 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 \[ \frac{x}{a}-\frac{\log \left (a+b e^{m x}\right )}{a m} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^(m*x))^(-1),x]

[Out]

x/a - Log[a + b*E^(m*x)]/(a*m)

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Rubi in Sympy [A]  time = 3.01905, size = 22, normalized size = 0.92 \[ - \frac{\log{\left (a + b e^{m x} \right )}}{a m} + \frac{\log{\left (e^{m x} \right )}}{a m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00689755, size = 24, normalized size = 1. \[ \frac{x}{a}-\frac{\log \left (a+b e^{m x}\right )}{a m} \]

Antiderivative was successfully verified.

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

[Out]

x/a - Log[a + b*E^(m*x)]/(a*m)

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Maple [A]  time = 0.003, size = 31, normalized size = 1.3 \[{\frac{\ln \left ({{\rm e}^{mx}} \right ) }{ma}}-{\frac{\ln \left ( a+b{{\rm e}^{mx}} \right ) }{ma}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*exp(m*x)),x)

[Out]

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

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Maxima [A]  time = 1.43155, size = 31, normalized size = 1.29 \[ \frac{x}{a} - \frac{\log \left (b e^{\left (m x\right )} + a\right )}{a m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/a - log(b*e^(m*x) + a)/(a*m)

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Fricas [A]  time = 0.20871, size = 30, normalized size = 1.25 \[ \frac{m x - \log \left (b e^{\left (m x\right )} + a\right )}{a m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(m*x - log(b*e^(m*x) + a))/(a*m)

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Sympy [A]  time = 0.125135, size = 15, normalized size = 0.62 \[ \frac{x}{a} - \frac{\log{\left (\frac{a}{b} + e^{m x} \right )}}{a m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*exp(m*x)),x)

[Out]

x/a - log(a/b + exp(m*x))/(a*m)

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GIAC/XCAS [A]  time = 0.199144, size = 32, normalized size = 1.33 \[ \frac{x}{a} - \frac{{\rm ln}\left ({\left | b e^{\left (m x\right )} + a \right |}\right )}{a m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/a - ln(abs(b*e^(m*x) + a))/(a*m)