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

Optimal. Leaf size=34 \[ -2 \text{PolyLog}\left (2,-e^x\right )-\frac{x^2}{e^x+1}+x^2-2 x \log \left (e^x+1\right ) \]

[Out]

x^2 - x^2/(1 + E^x) - 2*x*Log[1 + E^x] - 2*PolyLog[2, -E^x]

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Rubi [A]  time = 0.37813, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -2 \text{PolyLog}\left (2,-e^x\right )-\frac{x^2}{e^x+1}+x^2-2 x \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + E^(-x) + E^x),x]

[Out]

x^2 - x^2/(1 + E^x) - 2*x*Log[1 + E^x] - 2*PolyLog[2, -E^x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0565122, size = 33, normalized size = 0.97 \[ x \left (\frac{e^x x}{e^x+1}-2 \log \left (e^x+1\right )\right )-2 \text{PolyLog}\left (2,-e^x\right ) \]

Antiderivative was successfully verified.

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

[Out]

x*((E^x*x)/(1 + E^x) - 2*Log[1 + E^x]) - 2*PolyLog[2, -E^x]

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Maple [A]  time = 0.031, size = 32, normalized size = 0.9 \[{x}^{2}-{\frac{{x}^{2}}{1+{{\rm e}^{x}}}}-2\,x\ln \left ( 1+{{\rm e}^{x}} \right ) -2\,{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(2+exp(-x)+exp(x)),x)

[Out]

x^2-x^2/(1+exp(x))-2*x*ln(1+exp(x))-2*polylog(2,-exp(x))

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Maxima [A]  time = 0.81497, size = 41, normalized size = 1.21 \[ x^{2} - 2 \, x \log \left (e^{x} + 1\right ) - \frac{x^{2}}{e^{x} + 1} - 2 \,{\rm Li}_2\left (-e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^2 - 2*x*log(e^x + 1) - x^2/(e^x + 1) - 2*dilog(-e^x)

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Fricas [A]  time = 0.257606, size = 51, normalized size = 1.5 \[ \frac{x^{2} e^{x} - 2 \,{\left (e^{x} + 1\right )}{\rm Li}_2\left (-e^{x}\right ) - 2 \,{\left (x e^{x} + x\right )} \log \left (e^{x} + 1\right )}{e^{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^2*e^x - 2*(e^x + 1)*dilog(-e^x) - 2*(x*e^x + x)*log(e^x + 1))/(e^x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{e^{\left (-x\right )} + e^{x} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(e^(-x) + e^x + 2), x)