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*Sqr
t[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*Sqrt[3])])/(
Sqrt[3]*(3*I - Sqrt[3])) + (4*PolyLog[3, (-2*E^x)/(3 - I*Sqrt[3])])/(Sqrt[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 = 0.308999, 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, Rules used = {2263, 2184, 2190, 2531, 2282, 6589} \[ -\frac{4 x \text{PolyLog}\left (2,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{4 x \text{PolyLog}\left (2,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{4 \text{PolyLog}\left (3,-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{4 \text{PolyLog}\left (3,-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )}+\frac{2 x^3}{3 \sqrt{3} \left (\sqrt{3}+3 i\right )}-\frac{2 x^3}{3 \sqrt{3} \left (-\sqrt{3}+3 i\right )}-\frac{2 x^2 \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+3 i\right )}+\frac{2 x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+3 i\right )} \]

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*Sqr
t[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*Sqrt[3])])/(
Sqrt[3]*(3*I - Sqrt[3])) + (4*PolyLog[3, (-2*E^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) - (4*PolyLog[3,
(-2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{x^2}{3+3 e^x+e^{2 x}} \, dx &=-\frac{(2 i) \int \frac{x^2}{3-i \sqrt{3}+2 e^x} \, dx}{\sqrt{3}}+\frac{(2 i) \int \frac{x^2}{3+i \sqrt{3}+2 e^x} \, dx}{\sqrt{3}}\\ &=-\frac{2 x^3}{3 \sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{2 x^3}{3 \sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{(4 i) \int \frac{e^x x^2}{3-i \sqrt{3}+2 e^x} \, dx}{\sqrt{3} \left (3-i \sqrt{3}\right )}-\frac{(4 i) \int \frac{e^x x^2}{3+i \sqrt{3}+2 e^x} \, dx}{\sqrt{3} \left (3+i \sqrt{3}\right )}\\ &=-\frac{2 x^3}{3 \sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{2 x^3}{3 \sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{2 x^2 \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{(4 i) \int x \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right ) \, dx}{\sqrt{3} \left (3-i \sqrt{3}\right )}+\frac{(4 i) \int x \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right ) \, dx}{\sqrt{3} \left (3+i \sqrt{3}\right )}\\ &=-\frac{2 x^3}{3 \sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{2 x^3}{3 \sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{2 x^2 \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{4 x \text{Li}_2\left (-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{4 x \text{Li}_2\left (-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{(4 i) \int \text{Li}_2\left (-\frac{2 e^x}{3-i \sqrt{3}}\right ) \, dx}{\sqrt{3} \left (3-i \sqrt{3}\right )}+\frac{(4 i) \int \text{Li}_2\left (-\frac{2 e^x}{3+i \sqrt{3}}\right ) \, dx}{\sqrt{3} \left (3+i \sqrt{3}\right )}\\ &=-\frac{2 x^3}{3 \sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{2 x^3}{3 \sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{2 x^2 \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{4 x \text{Li}_2\left (-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{4 x \text{Li}_2\left (-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{(4 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{2 i x}{3 i+\sqrt{3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt{3} \left (3-i \sqrt{3}\right )}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{2 i x}{-3 i+\sqrt{3}}\right )}{x} \, dx,x,e^x\right )}{\sqrt{3} \left (3+i \sqrt{3}\right )}\\ &=-\frac{2 x^3}{3 \sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{2 x^3}{3 \sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{2 x^2 \log \left (1+\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{2 x^2 \log \left (1+\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}-\frac{4 x \text{Li}_2\left (-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}+\frac{4 x \text{Li}_2\left (-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}+\frac{4 \text{Li}_3\left (-\frac{2 e^x}{3-i \sqrt{3}}\right )}{\sqrt{3} \left (3 i+\sqrt{3}\right )}-\frac{4 \text{Li}_3\left (-\frac{2 e^x}{3+i \sqrt{3}}\right )}{\sqrt{3} \left (3 i-\sqrt{3}\right )}\\ \end{align*}

Mathematica [A]  time = 0.165446, size = 216, normalized size = 0.74 \[ \frac{2 i \left (\frac{2 \left (x \text{PolyLog}\left (2,-\frac{1}{2} \left (3+i \sqrt{3}\right ) e^{-x}\right )+\text{PolyLog}\left (3,-\frac{1}{2} i \left (\sqrt{3}-3 i\right ) e^{-x}\right )\right )}{3+i \sqrt{3}}-\frac{2 i \left (x \text{PolyLog}\left (2,\frac{1}{2} i \left (\sqrt{3}+3 i\right ) e^{-x}\right )+\text{PolyLog}\left (3,\frac{1}{2} i \left (\sqrt{3}+3 i\right ) e^{-x}\right )\right )}{\sqrt{3}+3 i}+\frac{i x^2 \log \left (1+\frac{1}{2} \left (3-i \sqrt{3}\right ) e^{-x}\right )}{\sqrt{3}+3 i}+\frac{i x^2 \log \left (1+\frac{1}{2} \left (3+i \sqrt{3}\right ) e^{-x}\right )}{\sqrt{3}-3 i}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{3+3\,{{\rm e}^{x}}+{{\rm e}^{2\,x}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{x^{2}}{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^2/(3+3*exp(x)+exp(2*x)),x, algorithm="maxima")

[Out]

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

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Fricas [C]  time = 1.65442, size = 520, normalized size = 1.77 \begin{align*} \frac{1}{9} \, x^{3} + \frac{1}{18} \,{\left (6 i \, \sqrt{3} x - 6 \, x\right )}{\rm Li}_2\left (-\frac{1}{6} \,{\left (i \, \sqrt{3} + 3\right )} e^{x}\right ) + \frac{1}{18} \,{\left (-6 i \, \sqrt{3} x - 6 \, x\right )}{\rm Li}_2\left (-\frac{1}{6} \,{\left (-i \, \sqrt{3} + 3\right )} e^{x}\right ) + \frac{1}{18} \,{\left (3 i \, \sqrt{3} x^{2} - 3 \, x^{2}\right )} \log \left (\frac{1}{6} \,{\left (i \, \sqrt{3} + 3\right )} e^{x} + 1\right ) + \frac{1}{18} \,{\left (-3 i \, \sqrt{3} x^{2} - 3 \, x^{2}\right )} \log \left (\frac{1}{6} \,{\left (-i \, \sqrt{3} + 3\right )} e^{x} + 1\right ) - \frac{1}{3} \,{\left (-i \, \sqrt{3} - 1\right )}{\rm polylog}\left (3, \frac{1}{6} \,{\left (i \, \sqrt{3} - 3\right )} e^{x}\right ) - \frac{1}{3} \,{\left (i \, \sqrt{3} - 1\right )}{\rm polylog}\left (3, \frac{1}{6} \,{\left (-i \, \sqrt{3} - 3\right )} e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^3 + 1/18*(6*I*sqrt(3)*x - 6*x)*dilog(-1/6*(I*sqrt(3) + 3)*e^x) + 1/18*(-6*I*sqrt(3)*x - 6*x)*dilog(-1/6*
(-I*sqrt(3) + 3)*e^x) + 1/18*(3*I*sqrt(3)*x^2 - 3*x^2)*log(1/6*(I*sqrt(3) + 3)*e^x + 1) + 1/18*(-3*I*sqrt(3)*x
^2 - 3*x^2)*log(1/6*(-I*sqrt(3) + 3)*e^x + 1) - 1/3*(-I*sqrt(3) - 1)*polylog(3, 1/6*(I*sqrt(3) - 3)*e^x) - 1/3
*(I*sqrt(3) - 1)*polylog(3, 1/6*(-I*sqrt(3) - 3)*e^x)

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

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{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^2/(3+3*exp(x)+exp(2*x)),x, algorithm="giac")

[Out]

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