3.281 \(\int \frac{e^{2 x}}{1+e^x} \, dx\)

Optimal. Leaf size=12 \[ e^x-\log \left (e^x+1\right ) \]

[Out]

E^x - Log[1 + E^x]

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Rubi [A]  time = 0.0344052, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ e^x-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^(2*x)/(1 + E^x),x]

[Out]

E^x - Log[1 + E^x]

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Rubi in Sympy [A]  time = 3.34127, size = 8, normalized size = 0.67 \[ e^{x} - \log{\left (e^{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(2*x)/(1+exp(x)),x)

[Out]

exp(x) - log(exp(x) + 1)

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Mathematica [A]  time = 0.00543907, size = 12, normalized size = 1. \[ e^x-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(2*x)/(1 + E^x),x]

[Out]

E^x - Log[1 + E^x]

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Maple [A]  time = 0., size = 11, normalized size = 0.9 \[{{\rm e}^{x}}-\ln \left ( 1+{{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*x)/(1+exp(x)),x)

[Out]

exp(x)-ln(1+exp(x))

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Maxima [A]  time = 1.33294, size = 14, normalized size = 1.17 \[ e^{x} - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(e^x + 1),x, algorithm="maxima")

[Out]

e^x - log(e^x + 1)

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Fricas [A]  time = 0.211083, size = 14, normalized size = 1.17 \[ e^{x} - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(e^x + 1),x, algorithm="fricas")

[Out]

e^x - log(e^x + 1)

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Sympy [A]  time = 0.078676, size = 8, normalized size = 0.67 \[ e^{x} - \log{\left (e^{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*x)/(1+exp(x)),x)

[Out]

exp(x) - log(exp(x) + 1)

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GIAC/XCAS [A]  time = 0.229322, size = 14, normalized size = 1.17 \[ e^{x} -{\rm ln}\left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(e^x + 1),x, algorithm="giac")

[Out]

e^x - ln(e^x + 1)