3.530 \(\int \frac{x}{2+e^{-x}+e^x} \, dx\)

Optimal. Leaf size=20 \[ -\frac{x}{e^x+1}+x-\log \left (e^x+1\right ) \]

[Out]

x - x/(1 + E^x) - Log[1 + E^x]

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Rubi [A]  time = 0.204094, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{x}{e^x+1}+x-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(2 + E^(-x) + E^x),x]

[Out]

x - x/(1 + E^x) - Log[1 + E^x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x e^{x}}{e^{2 x} + 2 e^{x} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*exp(x)/(exp(2*x) + 2*exp(x) + 1), x)

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Mathematica [A]  time = 0.0184128, size = 20, normalized size = 1. \[ -\frac{x}{e^x+1}+x-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(2 + E^(-x) + E^x),x]

[Out]

x - x/(1 + E^x) - Log[1 + E^x]

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Maple [A]  time = 0.012, size = 19, normalized size = 1. \[ -\ln \left ( 1+{{\rm e}^{x}} \right ) +{\frac{x{{\rm e}^{x}}}{1+{{\rm e}^{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(2+exp(-x)+exp(x)),x)

[Out]

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

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Maxima [A]  time = 0.759634, size = 24, normalized size = 1.2 \[ \frac{x e^{x}}{e^{x} + 1} - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^x/(e^x + 1) - log(e^x + 1)

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Fricas [A]  time = 0.245383, size = 31, normalized size = 1.55 \[ \frac{x e^{x} -{\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right )}{e^{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x*e^x - (e^x + 1)*log(e^x + 1))/(e^x + 1)

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Sympy [A]  time = 0.076257, size = 14, normalized size = 0.7 \[ x - \frac{x}{e^{x} + 1} - \log{\left (e^{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - x/(exp(x) + 1) - log(exp(x) + 1)

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GIAC/XCAS [A]  time = 0.246447, size = 38, normalized size = 1.9 \[ \frac{x e^{x} - e^{x}{\rm ln}\left (e^{x} + 1\right ) -{\rm ln}\left (e^{x} + 1\right )}{e^{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x*e^x - e^x*ln(e^x + 1) - ln(e^x + 1))/(e^x + 1)