3.40 \(\int \frac{e^x x}{1-e^{2 x}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2} \text{PolyLog}\left (2,-e^x\right )-\frac{1}{2} \text{PolyLog}\left (2,e^x\right )+x \tanh ^{-1}\left (e^x\right ) \]

[Out]

x*ArcTanh[E^x] + PolyLog[2, -E^x]/2 - PolyLog[2, E^x]/2

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Rubi [A]  time = 0.088203, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{1}{2} \text{PolyLog}\left (2,-e^x\right )-\frac{1}{2} \text{PolyLog}\left (2,e^x\right )+x \tanh ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(E^x*x)/(1 - E^(2*x)),x]

[Out]

x*ArcTanh[E^x] + PolyLog[2, -E^x]/2 - PolyLog[2, E^x]/2

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Rubi in Sympy [A]  time = 17.6102, size = 20, normalized size = 0.74 \[ x \operatorname{atanh}{\left (e^{x} \right )} + \frac{\operatorname{Li}_{2}\left (- e^{x}\right )}{2} - \frac{\operatorname{Li}_{2}\left (e^{x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*atanh(exp(x)) + polylog(2, -exp(x))/2 - polylog(2, exp(x))/2

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Mathematica [A]  time = 0.0234519, size = 38, normalized size = 1.41 \[ \frac{1}{2} \left (\text{PolyLog}\left (2,-e^x\right )-\text{PolyLog}\left (2,e^x\right )+x \left (\log \left (e^x+1\right )-\log \left (1-e^x\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(E^x*x)/(1 - E^(2*x)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 34, normalized size = 1.3 \[{\frac{x\ln \left ( 1+{{\rm e}^{x}} \right ) }{2}}+{\frac{{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) }{2}}-{\frac{x\ln \left ( 1-{{\rm e}^{x}} \right ) }{2}}-{\frac{{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*x/(1-exp(2*x)),x)

[Out]

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

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Maxima [A]  time = 0.83287, size = 42, normalized size = 1.56 \[ \frac{1}{2} \, x \log \left (e^{x} + 1\right ) - \frac{1}{2} \, x \log \left (-e^{x} + 1\right ) + \frac{1}{2} \,{\rm Li}_2\left (-e^{x}\right ) - \frac{1}{2} \,{\rm Li}_2\left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.252286, size = 42, normalized size = 1.56 \[ \frac{1}{2} \, x \log \left (e^{x} + 1\right ) - \frac{1}{2} \, x \log \left (-e^{x} + 1\right ) + \frac{1}{2} \,{\rm Li}_2\left (-e^{x}\right ) - \frac{1}{2} \,{\rm Li}_2\left (e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x*exp(x)/(exp(2*x) - 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x e^{x}}{e^{\left (2 \, x\right )} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-x*e^x/(e^(2*x) - 1), x)