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

Optimal. Leaf size=72 \[ -2 x \text{PolyLog}\left (2,-e^x\right )+2 \text{PolyLog}\left (2,-e^x\right )+2 \text{PolyLog}\left (3,-e^x\right )+\frac{x^3}{3}+\frac{x^2}{e^x+1}-x^2-x^2 \log \left (e^x+1\right )+2 x \log \left (e^x+1\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.229501, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {6688, 2185, 2184, 2190, 2531, 2282, 6589, 2191, 2279, 2391} \[ -2 x \text{PolyLog}\left (2,-e^x\right )+2 \text{PolyLog}\left (2,-e^x\right )+2 \text{PolyLog}\left (3,-e^x\right )+\frac{x^3}{3}+\frac{x^2}{e^x+1}-x^2-x^2 \log \left (e^x+1\right )+2 x \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(1 + 2*E^x + E^(2*x)),x]

[Out]

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 2185

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

Rule 2191

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

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

Mathematica [A]  time = 0.091594, size = 57, normalized size = 0.79 \[ -2 (x-1) \text{PolyLog}\left (2,-e^x\right )+2 \text{PolyLog}\left (3,-e^x\right )+\frac{\left (e^x (x-3)+x\right ) x^2}{3 \left (e^x+1\right )}-(x-2) x \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 65, normalized size = 0.9 \begin{align*} -{x}^{2}+{\frac{{x}^{2}}{1+{{\rm e}^{x}}}}+{\frac{{x}^{3}}{3}}+2\,x\ln \left ( 1+{{\rm e}^{x}} \right ) -{x}^{2}\ln \left ( 1+{{\rm e}^{x}} \right ) +2\,{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) -2\,x{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) +2\,{\it polylog} \left ( 3,-{{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(1+2*exp(x)+exp(2*x)),x)

[Out]

-x^2+x^2/(1+exp(x))+1/3*x^3+2*x*ln(1+exp(x))-x^2*ln(1+exp(x))+2*polylog(2,-exp(x))-2*x*polylog(2,-exp(x))+2*po
lylog(3,-exp(x))

________________________________________________________________________________________

Maxima [A]  time = 0.985644, size = 84, normalized size = 1.17 \begin{align*} \frac{1}{3} \, x^{3} - x^{2} \log \left (e^{x} + 1\right ) - x^{2} - 2 \, x{\rm Li}_2\left (-e^{x}\right ) + 2 \, x \log \left (e^{x} + 1\right ) + \frac{x^{2}}{e^{x} + 1} + 2 \,{\rm Li}_2\left (-e^{x}\right ) + 2 \,{\rm Li}_{3}(-e^{x}) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+2*exp(x)+exp(2*x)),x, algorithm="maxima")

[Out]

1/3*x^3 - x^2*log(e^x + 1) - x^2 - 2*x*dilog(-e^x) + 2*x*log(e^x + 1) + x^2/(e^x + 1) + 2*dilog(-e^x) + 2*poly
log(3, -e^x)

________________________________________________________________________________________

Fricas [C]  time = 1.49997, size = 212, normalized size = 2.94 \begin{align*} \frac{x^{3} - 6 \,{\left ({\left (x - 1\right )} e^{x} + x - 1\right )}{\rm Li}_2\left (-e^{x}\right ) +{\left (x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \,{\left (x^{2} +{\left (x^{2} - 2 \, x\right )} e^{x} - 2 \, x\right )} \log \left (e^{x} + 1\right ) + 6 \,{\left (e^{x} + 1\right )}{\rm polylog}\left (3, -e^{x}\right )}{3 \,{\left (e^{x} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+2*exp(x)+exp(2*x)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{2}}{e^{x} + 1} + \int \frac{x \left (x - 2\right )}{e^{x} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(1+2*exp(x)+exp(2*x)),x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{e^{\left (2 \, x\right )} + 2 \, e^{x} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+2*exp(x)+exp(2*x)),x, algorithm="giac")

[Out]

integrate(x^2/(e^(2*x) + 2*e^x + 1), x)