3.6.18 \(\int \frac {x^2}{3+3 e^x+e^{2 x}} \, dx\) [518]

Optimal. Leaf size=293 \[ -\frac {2 x^3}{3 \sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {2 x^3}{3 \sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {2 x^2 \log \left (1+\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {2 x^2 \log \left (1+\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {4 x \text {Li}_2\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {4 \text {Li}_3\left (-\frac {2 e^x}{3-i \sqrt {3}}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right )}-\frac {4 \text {Li}_3\left (-\frac {2 e^x}{3+i \sqrt {3}}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right )} \]

[Out]

-2/9*x^3/(3*I-3^(1/2))*3^(1/2)+2/3*x^2*ln(1+2*exp(x)/(3+I*3^(1/2)))/(3*I-3^(1/2))*3^(1/2)+4/3*x*polylog(2,-2*e
xp(x)/(3+I*3^(1/2)))/(3*I-3^(1/2))*3^(1/2)-4/3*polylog(3,-2*exp(x)/(3+I*3^(1/2)))/(3*I-3^(1/2))*3^(1/2)+2/9*x^
3*3^(1/2)/(3*I+3^(1/2))-2/3*x^2*ln(1+2*exp(x)/(3-I*3^(1/2)))*3^(1/2)/(3*I+3^(1/2))-4/3*x*polylog(2,-2*exp(x)/(
3-I*3^(1/2)))*3^(1/2)/(3*I+3^(1/2))+4/3*polylog(3,-2*exp(x)/(3-I*3^(1/2)))*3^(1/2)/(3*I+3^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 293, normalized size of antiderivative = 1.00, 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 = {2295, 2215, 2221, 2611, 2320, 6724} \begin {gather*} -\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 )} \end {gather*}

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 2215

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2295

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 2320

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 2611

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 6724

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) \text {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) \text {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 = 0.14, size = 216, normalized size = 0.74 \begin {gather*} \frac {2 i \left (\frac {i x^2 \log \left (1+\frac {1}{2} \left (3-i \sqrt {3}\right ) e^{-x}\right )}{3 i+\sqrt {3}}+\frac {i x^2 \log \left (1+\frac {1}{2} \left (3+i \sqrt {3}\right ) e^{-x}\right )}{-3 i+\sqrt {3}}+\frac {2 \left (x \text {Li}_2\left (-\frac {1}{2} i \left (-3 i+\sqrt {3}\right ) e^{-x}\right )+\text {Li}_3\left (-\frac {1}{2} i \left (-3 i+\sqrt {3}\right ) e^{-x}\right )\right )}{3+i \sqrt {3}}-\frac {2 i \left (x \text {Li}_2\left (\frac {1}{2} i \left (3 i+\sqrt {3}\right ) e^{-x}\right )+\text {Li}_3\left (\frac {1}{2} i \left (3 i+\sqrt {3}\right ) e^{-x}\right )\right )}{3 i+\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

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

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{3+3 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}}\, dx\]

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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 150, normalized size = 0.51 \begin {gather*} \frac {1}{9} \, x^{3} - \frac {1}{3} \, {\left (-i \, \sqrt {3} x + x\right )} {\rm Li}_2\left (-\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x}\right ) - \frac {1}{3} \, {\left (i \, \sqrt {3} x + x\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^{2} + x^{2}\right )} \log \left (\frac {1}{6} \, {\left (i \, \sqrt {3} + 3\right )} e^{x} + 1\right ) - \frac {1}{6} \, {\left (i \, \sqrt {3} x^{2} + 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 {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{e^{2 x} + 3 e^{x} + 3}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(exp(2*x) + 3*exp(x) + 3),x)

[Out]

int(x^2/(exp(2*x) + 3*exp(x) + 3), x)

________________________________________________________________________________________