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

Optimal. Leaf size=180 \[ -\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{1-\sqrt{5}}\right )}{\sqrt{5} \left (1-\sqrt{5}\right )}+\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{1+\sqrt{5}}\right )}{\sqrt{5} \left (1+\sqrt{5}\right )}-\frac{x^2}{\sqrt{5} \left (1+\sqrt{5}\right )}+\frac{x^2}{\sqrt{5} \left (1-\sqrt{5}\right )}-\frac{2 x \log \left (\frac{2 e^x}{1-\sqrt{5}}+1\right )}{\sqrt{5} \left (1-\sqrt{5}\right )}+\frac{2 x \log \left (\frac{2 e^x}{1+\sqrt{5}}+1\right )}{\sqrt{5} \left (1+\sqrt{5}\right )} \]

[Out]

x^2/(Sqrt[5]*(1 - Sqrt[5])) - x^2/(Sqrt[5]*(1 + Sqrt[5])) - (2*x*Log[1 + (2*E^x)
/(1 - Sqrt[5])])/(Sqrt[5]*(1 - Sqrt[5])) + (2*x*Log[1 + (2*E^x)/(1 + Sqrt[5])])/
(Sqrt[5]*(1 + Sqrt[5])) - (2*PolyLog[2, (-2*E^x)/(1 - Sqrt[5])])/(Sqrt[5]*(1 - S
qrt[5])) + (2*PolyLog[2, (-2*E^x)/(1 + Sqrt[5])])/(Sqrt[5]*(1 + Sqrt[5]))

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Rubi [A]  time = 0.296935, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{1-\sqrt{5}}\right )}{\sqrt{5} \left (1-\sqrt{5}\right )}+\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{1+\sqrt{5}}\right )}{\sqrt{5} \left (1+\sqrt{5}\right )}-\frac{x^2}{\sqrt{5} \left (1+\sqrt{5}\right )}+\frac{x^2}{\sqrt{5} \left (1-\sqrt{5}\right )}-\frac{2 x \log \left (\frac{2 e^x}{1-\sqrt{5}}+1\right )}{\sqrt{5} \left (1-\sqrt{5}\right )}+\frac{2 x \log \left (\frac{2 e^x}{1+\sqrt{5}}+1\right )}{\sqrt{5} \left (1+\sqrt{5}\right )} \]

Antiderivative was successfully verified.

[In]  Int[x/(-1 + E^x + E^(2*x)),x]

[Out]

x^2/(Sqrt[5]*(1 - Sqrt[5])) - x^2/(Sqrt[5]*(1 + Sqrt[5])) - (2*x*Log[1 + (2*E^x)
/(1 - Sqrt[5])])/(Sqrt[5]*(1 - Sqrt[5])) + (2*x*Log[1 + (2*E^x)/(1 + Sqrt[5])])/
(Sqrt[5]*(1 + Sqrt[5])) - (2*PolyLog[2, (-2*E^x)/(1 - Sqrt[5])])/(Sqrt[5]*(1 - S
qrt[5])) + (2*PolyLog[2, (-2*E^x)/(1 + Sqrt[5])])/(Sqrt[5]*(1 + Sqrt[5]))

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Rubi in Sympy [A]  time = 18.7373, size = 129, normalized size = 0.72 \[ \frac{2 \sqrt{5} x \log{\left (1 + \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) e^{- x} \right )}}{5 \left (1 + \sqrt{5}\right )} - \frac{2 \sqrt{5} x \log{\left (1 + \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) e^{- x} \right )}}{5 \left (- \sqrt{5} + 1\right )} + \frac{2 \sqrt{5} \operatorname{Li}_{2}\left (\left (- \frac{1}{2} + \frac{\sqrt{5}}{2}\right ) e^{- x}\right )}{5 \left (- \sqrt{5} + 1\right )} - \frac{2 \sqrt{5} \operatorname{Li}_{2}\left (\left (- \frac{\sqrt{5}}{2} - \frac{1}{2}\right ) e^{- x}\right )}{5 \left (1 + \sqrt{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*x*log(1 + (1/2 + sqrt(5)/2)*exp(-x))/(5*(1 + sqrt(5))) - 2*sqrt(5)*x*l
og(1 + (-sqrt(5)/2 + 1/2)*exp(-x))/(5*(-sqrt(5) + 1)) + 2*sqrt(5)*polylog(2, (-1
/2 + sqrt(5)/2)*exp(-x))/(5*(-sqrt(5) + 1)) - 2*sqrt(5)*polylog(2, (-sqrt(5)/2 -
1/2)*exp(-x))/(5*(1 + sqrt(5)))

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Mathematica [A]  time = 0.286867, size = 119, normalized size = 0.66 \[ \frac{\left (1+\sqrt{5}\right ) \text{PolyLog}\left (2,\frac{2 e^x}{\sqrt{5}-1}\right )+\left (\sqrt{5}-1\right ) \text{PolyLog}\left (2,-\frac{2 e^x}{1+\sqrt{5}}\right )+x \left (-\sqrt{5} x+\left (1+\sqrt{5}\right ) \log \left (1-\frac{2 e^x}{\sqrt{5}-1}\right )+\left (\sqrt{5}-1\right ) \log \left (\frac{2 e^x}{1+\sqrt{5}}+1\right )\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-(Sqrt[5]*x) + (1 + Sqrt[5])*Log[1 - (2*E^x)/(-1 + Sqrt[5])] + (-1 + Sqrt[5]
)*Log[1 + (2*E^x)/(1 + Sqrt[5])]) + (1 + Sqrt[5])*PolyLog[2, (2*E^x)/(-1 + Sqrt[
5])] + (-1 + Sqrt[5])*PolyLog[2, (-2*E^x)/(1 + Sqrt[5])])/(2*Sqrt[5])

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Maple [C]  time = 0.018, size = 183, normalized size = 1. \[{\frac{\sqrt{5}x}{10}\ln \left ({\frac{\sqrt{5}-1-2\,{{\rm e}^{x}}}{\sqrt{5}-1}} \right ) }+{\frac{x}{2}\ln \left ({\frac{\sqrt{5}-1-2\,{{\rm e}^{x}}}{\sqrt{5}-1}} \right ) }-{\frac{\sqrt{5}x}{10}\ln \left ({\frac{1+2\,{{\rm e}^{x}}+\sqrt{5}}{\sqrt{5}+1}} \right ) }+{\frac{x}{2}\ln \left ({\frac{1+2\,{{\rm e}^{x}}+\sqrt{5}}{\sqrt{5}+1}} \right ) }+{\frac{\sqrt{5}}{10}{\it dilog} \left ({\frac{\sqrt{5}-1-2\,{{\rm e}^{x}}}{\sqrt{5}-1}} \right ) }+{\frac{1}{2}{\it dilog} \left ({\frac{\sqrt{5}-1-2\,{{\rm e}^{x}}}{\sqrt{5}-1}} \right ) }-{\frac{\sqrt{5}}{10}{\it dilog} \left ({\frac{1+2\,{{\rm e}^{x}}+\sqrt{5}}{\sqrt{5}+1}} \right ) }+{\frac{1}{2}{\it dilog} \left ({\frac{1+2\,{{\rm e}^{x}}+\sqrt{5}}{\sqrt{5}+1}} \right ) }-{\frac{{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-1+exp(x)+exp(2*x)),x)

[Out]

1/10*5^(1/2)*x*ln((5^(1/2)-1-2*exp(x))/(5^(1/2)-1))+1/2*x*ln((5^(1/2)-1-2*exp(x)
)/(5^(1/2)-1))-1/10*5^(1/2)*x*ln((1+2*exp(x)+5^(1/2))/(5^(1/2)+1))+1/2*x*ln((1+2
*exp(x)+5^(1/2))/(5^(1/2)+1))+1/10*5^(1/2)*dilog((5^(1/2)-1-2*exp(x))/(5^(1/2)-1
))+1/2*dilog((5^(1/2)-1-2*exp(x))/(5^(1/2)-1))-1/10*5^(1/2)*dilog((1+2*exp(x)+5^
(1/2))/(5^(1/2)+1))+1/2*dilog((1+2*exp(x)+5^(1/2))/(5^(1/2)+1))-1/2*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(e^(2*x) + e^x - 1), x)

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Fricas [A]  time = 0.256947, size = 162, normalized size = 0.9 \[ -\frac{1}{2} \, x^{2} - \frac{1}{10} \,{\left (\sqrt{5} - 5\right )}{\rm Li}_2\left (-\frac{\sqrt{5} + 2 \, e^{x} + 1}{\sqrt{5} + 1} + 1\right ) + \frac{1}{10} \,{\left (\sqrt{5} + 5\right )}{\rm Li}_2\left (-\frac{\sqrt{5} - 2 \, e^{x} - 1}{\sqrt{5} - 1} + 1\right ) - \frac{1}{10} \,{\left (\sqrt{5} x - 5 \, x\right )} \log \left (\frac{\sqrt{5} + 2 \, e^{x} + 1}{\sqrt{5} + 1}\right ) + \frac{1}{10} \,{\left (\sqrt{5} x + 5 \, x\right )} \log \left (\frac{\sqrt{5} - 2 \, e^{x} - 1}{\sqrt{5} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x^2 - 1/10*(sqrt(5) - 5)*dilog(-(sqrt(5) + 2*e^x + 1)/(sqrt(5) + 1) + 1) +
1/10*(sqrt(5) + 5)*dilog(-(sqrt(5) - 2*e^x - 1)/(sqrt(5) - 1) + 1) - 1/10*(sqrt(
5)*x - 5*x)*log((sqrt(5) + 2*e^x + 1)/(sqrt(5) + 1)) + 1/10*(sqrt(5)*x + 5*x)*lo
g((sqrt(5) - 2*e^x - 1)/(sqrt(5) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(e^(2*x) + e^x - 1), x)