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

Optimal. Leaf size=204 \[ -\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{x^2}{\sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{x^2}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}-\frac{2 x \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{2 x \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )} \]

[Out]

-(x^2/(Sqrt[3]*(3*I - Sqrt[3]))) + x^2/(Sqrt[3]*(3*I + Sqrt[3])) - (2*x*Log[1 + (2*E^x)/(3 - I*Sqrt[3])])/(Sqr
t[3]*(3*I + Sqrt[3])) + (2*x*Log[1 + (2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) - (2*PolyLog[2, (-2*E
^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*PolyLog[2, (-2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqr
t[3]))

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Rubi [A]  time = 0.195367, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {2263, 2184, 2190, 2279, 2391} \[ -\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{2 \text{PolyLog}\left (2,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{x^2}{\sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{x^2}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}-\frac{2 x \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{2 x \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(3 + 3*E^x + E^(2*x)),x]

[Out]

-(x^2/(Sqrt[3]*(3*I - Sqrt[3]))) + x^2/(Sqrt[3]*(3*I + Sqrt[3])) - (2*x*Log[1 + (2*E^x)/(3 - I*Sqrt[3])])/(Sqr
t[3]*(3*I + Sqrt[3])) + (2*x*Log[1 + (2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) - (2*PolyLog[2, (-2*E
^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*PolyLog[2, (-2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqr
t[3]))

Rule 2263

Int[((f_.) + (g_.)*(x_))^(m_.)/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*
a*c, 2]}, Dist[(2*c)/q, Int[(f + g*x)^m/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[(f + g*x)^m/(b + q + 2*c
*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[
m, 0]

Rule 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x}{3+3 e^x+e^{2 x}} \, dx &=-\frac{(2 i) \int \frac{x}{3-i \sqrt{3}+2 e^x} \, dx}{\sqrt{3}}+\frac{(2 i) \int \frac{x}{3+i \sqrt{3}+2 e^x} \, dx}{\sqrt{3}}\\ &=-\frac{x^2}{\sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{x^2}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{(4 i) \int \frac{e^x x}{3-i \sqrt{3}+2 e^x} \, dx}{\sqrt{3} \left (3-i \sqrt{3}\right )}-\frac{(4 i) \int \frac{e^x x}{3+i \sqrt{3}+2 e^x} \, dx}{\sqrt{3} \left (3+i \sqrt{3}\right )}\\ &=-\frac{x^2}{\sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{x^2}{\sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{2 x \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 x \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{(2 i) \int \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right ) \, dx}{\sqrt{3} \left (3-i \sqrt{3}\right )}+\frac{(2 i) \int \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right ) \, dx}{\sqrt{3} \left (3+i \sqrt{3}\right )}\\ &=-\frac{x^2}{\sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{x^2}{\sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{2 x \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 x \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{3-i \sqrt{3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt{3} \left (3-i \sqrt{3}\right )}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{3+i \sqrt{3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt{3} \left (3+i \sqrt{3}\right )}\\ &=-\frac{x^2}{\sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{x^2}{\sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{2 x \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 x \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{2 \text{Li}_2\left (-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 \text{Li}_2\left (-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}\\ \end{align*}

Mathematica [A]  time = 0.0902245, size = 144, normalized size = 0.71 \[ \frac{\left (\sqrt{3}+3 i\right ) \text{PolyLog}\left (2,-\frac{1}{2} \left (3+i \sqrt{3}\right ) e^{-x}\right )+\left (\sqrt{3}-3 i\right ) \text{PolyLog}\left (2,\frac{1}{2} i \left (\sqrt{3}+3 i\right ) e^{-x}\right )-x \left (\left (\sqrt{3}-3 i\right ) \log \left (1+\frac{1}{2} \left (3-i \sqrt{3}\right ) e^{-x}\right )+\left (\sqrt{3}+3 i\right ) \log \left (1+\frac{1}{2} \left (3+i \sqrt{3}\right ) e^{-x}\right )\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(3 + 3*E^x + E^(2*x)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 235, normalized size = 1.2 \begin{align*}{\frac{i}{6}}\sqrt{3}x\ln \left ({\frac{i\sqrt{3}-2\,{{\rm e}^{x}}-3}{i\sqrt{3}-3}} \right ) -{\frac{x}{6}\ln \left ({\frac{i\sqrt{3}-2\,{{\rm e}^{x}}-3}{i\sqrt{3}-3}} \right ) }-{\frac{i}{6}}\sqrt{3}x\ln \left ({\frac{i\sqrt{3}+2\,{{\rm e}^{x}}+3}{3+i\sqrt{3}}} \right ) -{\frac{x}{6}\ln \left ({\frac{i\sqrt{3}+2\,{{\rm e}^{x}}+3}{3+i\sqrt{3}}} \right ) }+{\frac{i}{6}}\sqrt{3}{\it dilog} \left ({\frac{i\sqrt{3}-2\,{{\rm e}^{x}}-3}{i\sqrt{3}-3}} \right ) -{\frac{1}{6}{\it dilog} \left ({\frac{i\sqrt{3}-2\,{{\rm e}^{x}}-3}{i\sqrt{3}-3}} \right ) }-{\frac{i}{6}}\sqrt{3}{\it dilog} \left ({\frac{i\sqrt{3}+2\,{{\rm e}^{x}}+3}{3+i\sqrt{3}}} \right ) -{\frac{1}{6}{\it dilog} \left ({\frac{i\sqrt{3}+2\,{{\rm e}^{x}}+3}{3+i\sqrt{3}}} \right ) }+{\frac{{x}^{2}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*I*3^(1/2)*x*ln((I*3^(1/2)-2*exp(x)-3)/(I*3^(1/2)-3))-1/6*x*ln((I*3^(1/2)-2*exp(x)-3)/(I*3^(1/2)-3))-1/6*I*
3^(1/2)*x*ln((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))-1/6*x*ln((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))+1/6*I*3^(1/2
)*dilog((I*3^(1/2)-2*exp(x)-3)/(I*3^(1/2)-3))-1/6*dilog((I*3^(1/2)-2*exp(x)-3)/(I*3^(1/2)-3))-1/6*I*3^(1/2)*di
log((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))-1/6*dilog((I*3^(1/2)+2*exp(x)+3)/(3+I*3^(1/2)))+1/6*x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3+3*exp(x)+exp(2*x)),x, algorithm="maxima")

[Out]

integrate(x/(e^(2*x) + 3*e^x + 3), x)

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Fricas [A]  time = 1.60891, size = 317, normalized size = 1.55 \begin{align*} \frac{1}{6} \, x^{2} + \frac{1}{6} \,{\left (i \, \sqrt{3} - 1\right )}{\rm Li}_2\left (-\frac{1}{6} \,{\left (i \, \sqrt{3} + 3\right )} e^{x}\right ) + \frac{1}{6} \,{\left (-i \, \sqrt{3} - 1\right )}{\rm Li}_2\left (-\frac{1}{6} \,{\left (-i \, \sqrt{3} + 3\right )} e^{x}\right ) + \frac{1}{6} \,{\left (i \, \sqrt{3} x - x\right )} \log \left (\frac{1}{6} \,{\left (i \, \sqrt{3} + 3\right )} e^{x} + 1\right ) + \frac{1}{6} \,{\left (-i \, \sqrt{3} x - x\right )} \log \left (\frac{1}{6} \,{\left (-i \, \sqrt{3} + 3\right )} e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3+3*exp(x)+exp(2*x)),x, algorithm="fricas")

[Out]

1/6*x^2 + 1/6*(I*sqrt(3) - 1)*dilog(-1/6*(I*sqrt(3) + 3)*e^x) + 1/6*(-I*sqrt(3) - 1)*dilog(-1/6*(-I*sqrt(3) +
3)*e^x) + 1/6*(I*sqrt(3)*x - x)*log(1/6*(I*sqrt(3) + 3)*e^x + 1) + 1/6*(-I*sqrt(3)*x - x)*log(1/6*(-I*sqrt(3)
+ 3)*e^x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(3+3*exp(x)+exp(2*x)),x, algorithm="giac")

[Out]

integrate(x/(e^(2*x) + 3*e^x + 3), x)