3.2 \(\int \frac{e^x}{a+b e^x} \, dx\)

Optimal. Leaf size=12 \[ \frac{\log \left (a+b e^x\right )}{b} \]

[Out]

Log[a + b*E^x]/b

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Rubi [A]  time = 0.0339316, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\log \left (a+b e^x\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[E^x/(a + b*E^x),x]

[Out]

Log[a + b*E^x]/b

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Rubi in Sympy [A]  time = 6.93766, size = 8, normalized size = 0.67 \[ \frac{\log{\left (a + b e^{x} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*exp(x))/b

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Mathematica [A]  time = 0.00343598, size = 12, normalized size = 1. \[ \frac{\log \left (a+b e^x\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(a + b*E^x),x]

[Out]

Log[a + b*E^x]/b

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Maple [A]  time = 0.003, size = 12, normalized size = 1. \[{\frac{\ln \left ( a+b{{\rm e}^{x}} \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(a+b*exp(x)),x)

[Out]

ln(a+b*exp(x))/b

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Maxima [A]  time = 0.794781, size = 15, normalized size = 1.25 \[ \frac{\log \left (b e^{x} + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*e^x + a)/b

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Fricas [A]  time = 0.296313, size = 15, normalized size = 1.25 \[ \frac{\log \left (b e^{x} + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*e^x + a)/b

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Sympy [A]  time = 0.201134, size = 8, normalized size = 0.67 \[ \frac{\log{\left (\frac{a}{b} + e^{x} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(a+b*exp(x)),x)

[Out]

log(a/b + exp(x))/b

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GIAC/XCAS [A]  time = 0.235233, size = 16, normalized size = 1.33 \[ \frac{{\rm ln}\left ({\left | b e^{x} + a \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*e^x + a))/b