3.518 \(\int \frac{x^2}{3+3 e^x+e^{2 x}} \, dx\)

Optimal. Leaf size=293 \[ -\frac{4 x \text{PolyLog}\left (2,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{4 x \text{PolyLog}\left (2,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{4 \text{PolyLog}\left (3,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{4 \text{PolyLog}\left (3,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{2 x^3}{3 \sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{2 x^3}{3 \sqrt{3} \left (-\sqrt{3}+3 i\right )}-\frac{2 x^2 \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{2 x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )} \]

[Out]

(-2*x^3)/(3*Sqrt[3]*(3*I - Sqrt[3])) + (2*x^3)/(3*Sqrt[3]*(3*I + Sqrt[3])) - (2*
x^2*Log[1 + (2*E^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*x^2*Log[1 +
(2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) - (4*x*PolyLog[2, (-2*E^x)/
(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (4*x*PolyLog[2, (-2*E^x)/(3 + I*Sq
rt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) + (4*PolyLog[3, (-2*E^x)/(3 - I*Sqrt[3])])/(S
qrt[3]*(3*I + Sqrt[3])) - (4*PolyLog[3, (-2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I
- Sqrt[3]))

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Rubi [A]  time = 0.470417, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{4 x \text{PolyLog}\left (2,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{4 x \text{PolyLog}\left (2,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{4 \text{PolyLog}\left (3,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{4 \text{PolyLog}\left (3,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{2 x^3}{3 \sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{2 x^3}{3 \sqrt{3} \left (-\sqrt{3}+3 i\right )}-\frac{2 x^2 \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{2 x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(3 + 3*E^x + E^(2*x)),x]

[Out]

(-2*x^3)/(3*Sqrt[3]*(3*I - Sqrt[3])) + (2*x^3)/(3*Sqrt[3]*(3*I + Sqrt[3])) - (2*
x^2*Log[1 + (2*E^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*x^2*Log[1 +
(2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) - (4*x*PolyLog[2, (-2*E^x)/
(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (4*x*PolyLog[2, (-2*E^x)/(3 + I*Sq
rt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) + (4*PolyLog[3, (-2*E^x)/(3 - I*Sqrt[3])])/(S
qrt[3]*(3*I + Sqrt[3])) - (4*PolyLog[3, (-2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I
- Sqrt[3]))

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Rubi in Sympy [A]  time = 45.7132, size = 189, normalized size = 0.65 \[ - x^{2} \left (\frac{1}{6} - \frac{\sqrt{3} i}{6}\right ) \log{\left (1 + \left (\frac{3}{2} - \frac{\sqrt{3} i}{2}\right ) e^{- x} \right )} - x^{2} \left (\frac{1}{6} + \frac{\sqrt{3} i}{6}\right ) \log{\left (1 + \left (\frac{3}{2} + \frac{\sqrt{3} i}{2}\right ) e^{- x} \right )} + x \left (\frac{1}{3} + \frac{\sqrt{3} i}{3}\right ) \operatorname{Li}_{2}\left (\left (- \frac{3}{2} - \frac{\sqrt{3} i}{2}\right ) e^{- x}\right ) + x \left (\frac{1}{3} - \frac{\sqrt{3} i}{3}\right ) \operatorname{Li}_{2}\left (\left (- \frac{3}{2} + \frac{\sqrt{3} i}{2}\right ) e^{- x}\right ) + \left (\frac{1}{3} + \frac{\sqrt{3} i}{3}\right ) \operatorname{Li}_{3}\left (\left (- \frac{3}{2} - \frac{\sqrt{3} i}{2}\right ) e^{- x}\right ) + \left (\frac{1}{3} - \frac{\sqrt{3} i}{3}\right ) \operatorname{Li}_{3}\left (\left (- \frac{3}{2} + \frac{\sqrt{3} i}{2}\right ) e^{- x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(3+3*exp(x)+exp(2*x)),x)

[Out]

-x**2*(1/6 - sqrt(3)*I/6)*log(1 + (3/2 - sqrt(3)*I/2)*exp(-x)) - x**2*(1/6 + sqr
t(3)*I/6)*log(1 + (3/2 + sqrt(3)*I/2)*exp(-x)) + x*(1/3 + sqrt(3)*I/3)*polylog(2
, (-3/2 - sqrt(3)*I/2)*exp(-x)) + x*(1/3 - sqrt(3)*I/3)*polylog(2, (-3/2 + sqrt(
3)*I/2)*exp(-x)) + (1/3 + sqrt(3)*I/3)*polylog(3, (-3/2 - sqrt(3)*I/2)*exp(-x))
+ (1/3 - sqrt(3)*I/3)*polylog(3, (-3/2 + sqrt(3)*I/2)*exp(-x))

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Mathematica [A]  time = 0.135723, size = 304, normalized size = 1.04 \[ \frac{-6 \left (\sqrt{3}-3 i\right ) x \text{PolyLog}\left (2,\frac{2 e^x}{-3+i \sqrt{3}}\right )-6 \left (\sqrt{3}+3 i\right ) x \text{PolyLog}\left (2,-\frac{2 e^x}{3+i \sqrt{3}}\right )+6 \sqrt{3} \text{PolyLog}\left (3,\frac{2 e^x}{-3+i \sqrt{3}}\right )-18 i \text{PolyLog}\left (3,\frac{2 e^x}{-3+i \sqrt{3}}\right )+6 \sqrt{3} \text{PolyLog}\left (3,-\frac{2 e^x}{3+i \sqrt{3}}\right )+18 i \text{PolyLog}\left (3,-\frac{2 e^x}{3+i \sqrt{3}}\right )+2 \sqrt{3} x^3-3 \sqrt{3} x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )-9 i x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )-3 \sqrt{3} x^2 \log \left (1+\frac{2 i e^x}{\sqrt{3}+3 i}\right )+9 i x^2 \log \left (1+\frac{2 i e^x}{\sqrt{3}+3 i}\right )}{18 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(3 + 3*E^x + E^(2*x)),x]

[Out]

(2*Sqrt[3]*x^3 - (9*I)*x^2*Log[1 + (2*E^x)/(3 + I*Sqrt[3])] - 3*Sqrt[3]*x^2*Log[
1 + (2*E^x)/(3 + I*Sqrt[3])] + (9*I)*x^2*Log[1 + ((2*I)*E^x)/(3*I + Sqrt[3])] -
3*Sqrt[3]*x^2*Log[1 + ((2*I)*E^x)/(3*I + Sqrt[3])] - 6*(-3*I + Sqrt[3])*x*PolyLo
g[2, (2*E^x)/(-3 + I*Sqrt[3])] - 6*(3*I + Sqrt[3])*x*PolyLog[2, (-2*E^x)/(3 + I*
Sqrt[3])] - (18*I)*PolyLog[3, (2*E^x)/(-3 + I*Sqrt[3])] + 6*Sqrt[3]*PolyLog[3, (
2*E^x)/(-3 + I*Sqrt[3])] + (18*I)*PolyLog[3, (-2*E^x)/(3 + I*Sqrt[3])] + 6*Sqrt[
3]*PolyLog[3, (-2*E^x)/(3 + I*Sqrt[3])])/(18*Sqrt[3])

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{3+3\,{{\rm e}^{x}}+{{\rm e}^{2\,x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(3+3*exp(x)+exp(2*x)),x)

[Out]

int(x^2/(3+3*exp(x)+exp(2*x)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(e^(2*x) + 3*e^x + 3),x, algorithm="maxima")

[Out]

integrate(x^2/(e^(2*x) + 3*e^x + 3), x)

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Fricas [A]  time = 0.299708, size = 300, normalized size = 1.02 \[ \frac{1}{9} \, x^{3} + \frac{1}{18} \,{\left (-6 i \, \sqrt{3} x - 6 \, x\right )}{\rm Li}_2\left (-\frac{2 \, \sqrt{3} e^{x} + 3 \, \sqrt{3} + 3 i}{3 \, \sqrt{3} + 3 i} + 1\right ) + \frac{1}{18} \,{\left (6 i \, \sqrt{3} x - 6 \, x\right )}{\rm Li}_2\left (-\frac{2 \, \sqrt{3} e^{x} + 3 \, \sqrt{3} - 3 i}{3 \, \sqrt{3} - 3 i} + 1\right ) + \frac{1}{18} \,{\left (-3 i \, \sqrt{3} x^{2} - 3 \, x^{2}\right )} \log \left (\frac{2 \, \sqrt{3} e^{x} + 3 \, \sqrt{3} + 3 i}{3 \, \sqrt{3} + 3 i}\right ) + \frac{1}{18} \,{\left (3 i \, \sqrt{3} x^{2} - 3 \, x^{2}\right )} \log \left (\frac{2 \, \sqrt{3} e^{x} + 3 \, \sqrt{3} - 3 i}{3 \, \sqrt{3} - 3 i}\right ) - \frac{1}{3} \,{\left (-i \, \sqrt{3} - 1\right )}{\rm Li}_{3}(-\frac{2 \, \sqrt{3} e^{x}}{3 \, \sqrt{3} + 3 i}) - \frac{1}{3} \,{\left (i \, \sqrt{3} - 1\right )}{\rm Li}_{3}(-\frac{2 \, \sqrt{3} e^{x}}{3 \, \sqrt{3} - 3 i}) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(e^(2*x) + 3*e^x + 3),x, algorithm="fricas")

[Out]

1/9*x^3 + 1/18*(-6*I*sqrt(3)*x - 6*x)*dilog(-(2*sqrt(3)*e^x + 3*sqrt(3) + 3*I)/(
3*sqrt(3) + 3*I) + 1) + 1/18*(6*I*sqrt(3)*x - 6*x)*dilog(-(2*sqrt(3)*e^x + 3*sqr
t(3) - 3*I)/(3*sqrt(3) - 3*I) + 1) + 1/18*(-3*I*sqrt(3)*x^2 - 3*x^2)*log((2*sqrt
(3)*e^x + 3*sqrt(3) + 3*I)/(3*sqrt(3) + 3*I)) + 1/18*(3*I*sqrt(3)*x^2 - 3*x^2)*l
og((2*sqrt(3)*e^x + 3*sqrt(3) - 3*I)/(3*sqrt(3) - 3*I)) - 1/3*(-I*sqrt(3) - 1)*p
olylog(3, -2*sqrt(3)*e^x/(3*sqrt(3) + 3*I)) - 1/3*(I*sqrt(3) - 1)*polylog(3, -2*
sqrt(3)*e^x/(3*sqrt(3) - 3*I))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{e^{2 x} + 3 e^{x} + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(3+3*exp(x)+exp(2*x)),x)

[Out]

Integral(x**2/(exp(2*x) + 3*exp(x) + 3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{e^{\left (2 \, x\right )} + 3 \, e^{x} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(e^(2*x) + 3*e^x + 3),x, algorithm="giac")

[Out]

integrate(x^2/(e^(2*x) + 3*e^x + 3), x)