3.99 \(\int \log ^2(1+x+x^2) \, dx\)

Optimal. Leaf size=371 \[ -\left (1+i \sqrt{3}\right ) \text{PolyLog}\left (2,-\frac{2 i x-\sqrt{3}+i}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{PolyLog}\left (2,\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+x \log ^2\left (x^2+x+1\right )+\left (1-i \sqrt{3}\right ) \log \left (x^2+x+1\right ) \log \left (2 x-i \sqrt{3}+1\right )-4 x \log \left (x^2+x+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 x+i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )-2 \log \left (x^2+x+1\right )+8 x-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (2 x-i \sqrt{3}+1\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (2 x+i \sqrt{3}+1\right )-\left (1-i \sqrt{3}\right ) \log \left (-\frac{i \left (2 x+i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x-i \sqrt{3}+1\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (2 x-i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x+i \sqrt{3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

8*x - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - ((1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]^2)/2 - (1 + I*Sqrt[3])*L
og[((I/2)*(1 - I*Sqrt[3] + 2*x))/Sqrt[3]]*Log[1 + I*Sqrt[3] + 2*x] - ((1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*x]
^2)/2 - (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]*Log[((-I/2)*(1 + I*Sqrt[3] + 2*x))/Sqrt[3]] - 2*Log[1 + x + x
^2] - 4*x*Log[1 + x + x^2] + (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + (1 + I*Sqrt[3])*Log[1
 + I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + x*Log[1 + x + x^2]^2 - (1 + I*Sqrt[3])*PolyLog[2, -(I - Sqrt[3] + (2*I)
*x)/(2*Sqrt[3])] - (1 - I*Sqrt[3])*PolyLog[2, (I + Sqrt[3] + (2*I)*x)/(2*Sqrt[3])]

________________________________________________________________________________________

Rubi [A]  time = 0.541176, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 14, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.556, Rules used = {2523, 2528, 773, 634, 618, 204, 628, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\left (1+i \sqrt{3}\right ) \text{PolyLog}\left (2,-\frac{2 i x-\sqrt{3}+i}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{PolyLog}\left (2,\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+x \log ^2\left (x^2+x+1\right )+\left (1-i \sqrt{3}\right ) \log \left (x^2+x+1\right ) \log \left (2 x-i \sqrt{3}+1\right )-4 x \log \left (x^2+x+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 x+i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )-2 \log \left (x^2+x+1\right )+8 x-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (2 x-i \sqrt{3}+1\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (2 x+i \sqrt{3}+1\right )-\left (1-i \sqrt{3}\right ) \log \left (-\frac{i \left (2 x+i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x-i \sqrt{3}+1\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (2 x-i \sqrt{3}+1\right )}{2 \sqrt{3}}\right ) \log \left (2 x+i \sqrt{3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[1 + x + x^2]^2,x]

[Out]

8*x - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - ((1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]^2)/2 - (1 + I*Sqrt[3])*L
og[((I/2)*(1 - I*Sqrt[3] + 2*x))/Sqrt[3]]*Log[1 + I*Sqrt[3] + 2*x] - ((1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*x]
^2)/2 - (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]*Log[((-I/2)*(1 + I*Sqrt[3] + 2*x))/Sqrt[3]] - 2*Log[1 + x + x
^2] - 4*x*Log[1 + x + x^2] + (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + (1 + I*Sqrt[3])*Log[1
 + I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + x*Log[1 + x + x^2]^2 - (1 + I*Sqrt[3])*PolyLog[2, -(I - Sqrt[3] + (2*I)
*x)/(2*Sqrt[3])] - (1 - I*Sqrt[3])*PolyLog[2, (I + Sqrt[3] + (2*I)*x)/(2*Sqrt[3])]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
 d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
 b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
 && EqQ[e*f - d*g, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
 b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
 + c*(e*f - d*g), 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 \log ^2\left (1+x+x^2\right ) \, dx &=x \log ^2\left (1+x+x^2\right )-2 \int \frac{x (1+2 x) \log \left (1+x+x^2\right )}{1+x+x^2} \, dx\\ &=x \log ^2\left (1+x+x^2\right )-2 \int \left (2 \log \left (1+x+x^2\right )-\frac{(2+x) \log \left (1+x+x^2\right )}{1+x+x^2}\right ) \, dx\\ &=x \log ^2\left (1+x+x^2\right )+2 \int \frac{(2+x) \log \left (1+x+x^2\right )}{1+x+x^2} \, dx-4 \int \log \left (1+x+x^2\right ) \, dx\\ &=-4 x \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+2 \int \left (\frac{\left (1-i \sqrt{3}\right ) \log \left (1+x+x^2\right )}{1-i \sqrt{3}+2 x}+\frac{\left (1+i \sqrt{3}\right ) \log \left (1+x+x^2\right )}{1+i \sqrt{3}+2 x}\right ) \, dx+4 \int \frac{x (1+2 x)}{1+x+x^2} \, dx\\ &=8 x-4 x \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+4 \int \frac{-2-x}{1+x+x^2} \, dx+\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (1+x+x^2\right )}{1-i \sqrt{3}+2 x} \, dx+\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (1+x+x^2\right )}{1+i \sqrt{3}+2 x} \, dx\\ &=8 x-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-2 \int \frac{1+2 x}{1+x+x^2} \, dx-6 \int \frac{1}{1+x+x^2} \, dx+\left (-1-i \sqrt{3}\right ) \int \frac{(1+2 x) \log \left (1+i \sqrt{3}+2 x\right )}{1+x+x^2} \, dx+\left (-1+i \sqrt{3}\right ) \int \frac{(1+2 x) \log \left (1-i \sqrt{3}+2 x\right )}{1+x+x^2} \, dx\\ &=8 x-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+12 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )+\left (-1-i \sqrt{3}\right ) \int \left (\frac{2 \log \left (1+i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x}+\frac{2 \log \left (1+i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x}\right ) \, dx+\left (-1+i \sqrt{3}\right ) \int \left (\frac{2 \log \left (1-i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x}+\frac{2 \log \left (1-i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x}\right ) \, dx\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (1-i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x} \, dx-\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (1-i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x} \, dx-\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (1+i \sqrt{3}+2 x\right )}{1-i \sqrt{3}+2 x} \, dx-\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (1+i \sqrt{3}+2 x\right )}{1+i \sqrt{3}+2 x} \, dx\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 x\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-\left (1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i \sqrt{3}+2 x\right )+\left (2 \left (1-i \sqrt{3}\right )\right ) \int \frac{\log \left (\frac{2 \left (1+i \sqrt{3}+2 x\right )}{-2 \left (1-i \sqrt{3}\right )+2 \left (1+i \sqrt{3}\right )}\right )}{1-i \sqrt{3}+2 x} \, dx-\left (1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i \sqrt{3}+2 x\right )+\left (2 \left (1+i \sqrt{3}\right )\right ) \int \frac{\log \left (\frac{2 \left (1-i \sqrt{3}+2 x\right )}{2 \left (1-i \sqrt{3}\right )-2 \left (1+i \sqrt{3}\right )}\right )}{1+i \sqrt{3}+2 x} \, dx\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (1-i \sqrt{3}+2 x\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 x\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (1+i \sqrt{3}+2 x\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{-2 \left (1-i \sqrt{3}\right )+2 \left (1+i \sqrt{3}\right )}\right )}{x} \, dx,x,1-i \sqrt{3}+2 x\right )+\left (1+i \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{2 \left (1-i \sqrt{3}\right )-2 \left (1+i \sqrt{3}\right )}\right )}{x} \, dx,x,1+i \sqrt{3}+2 x\right )\\ &=8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-\frac{1}{2} \left (1-i \sqrt{3}\right ) \log ^2\left (1-i \sqrt{3}+2 x\right )-\left (1+i \sqrt{3}\right ) \log \left (\frac{i \left (1-i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right ) \log \left (1+i \sqrt{3}+2 x\right )-\frac{1}{2} \left (1+i \sqrt{3}\right ) \log ^2\left (1+i \sqrt{3}+2 x\right )-\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (-\frac{i \left (1+i \sqrt{3}+2 x\right )}{2 \sqrt{3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-\left (1+i \sqrt{3}\right ) \text{Li}_2\left (-\frac{i-\sqrt{3}+2 i x}{2 \sqrt{3}}\right )-\left (1-i \sqrt{3}\right ) \text{Li}_2\left (\frac{i+\sqrt{3}+2 i x}{2 \sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.146165, size = 323, normalized size = 0.87 \[ -\frac{1}{2} i \left (\sqrt{3}-i\right ) \left (2 \text{PolyLog}\left (2,\frac{-2 i x+\sqrt{3}-i}{2 \sqrt{3}}\right )+\log \left (2 x+i \sqrt{3}+1\right ) \left (2 \log \left (\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+\log \left (2 x+i \sqrt{3}+1\right )\right )\right )+\frac{1}{2} i \left (\sqrt{3}+i\right ) \left (2 \text{PolyLog}\left (2,\frac{2 i x+\sqrt{3}+i}{2 \sqrt{3}}\right )+\log \left (2 x-i \sqrt{3}+1\right ) \left (2 \log \left (\frac{-2 i x+\sqrt{3}-i}{2 \sqrt{3}}\right )+\log \left (2 x-i \sqrt{3}+1\right )\right )\right )+x \log ^2\left (x^2+x+1\right )-4 x \log \left (x^2+x+1\right )+\left (1-i \sqrt{3}\right ) \log \left (2 x-i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )+\left (1+i \sqrt{3}\right ) \log \left (2 x+i \sqrt{3}+1\right ) \log \left (x^2+x+1\right )-2 \log \left (x^2+x+1\right )+8 x-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + x + x^2]^2,x]

[Out]

8*x - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*Log[1 + x + x^2] - 4*x*Log[1 + x + x^2] + (1 - I*Sqrt[3])*Log[1
- I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + (1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + x*Log[1 + x
+ x^2]^2 - (I/2)*(-I + Sqrt[3])*(Log[1 + I*Sqrt[3] + 2*x]*(2*Log[(I + Sqrt[3] + (2*I)*x)/(2*Sqrt[3])] + Log[1
+ I*Sqrt[3] + 2*x]) + 2*PolyLog[2, (-I + Sqrt[3] - (2*I)*x)/(2*Sqrt[3])]) + (I/2)*(I + Sqrt[3])*(Log[1 - I*Sqr
t[3] + 2*x]*(2*Log[(-I + Sqrt[3] - (2*I)*x)/(2*Sqrt[3])] + Log[1 - I*Sqrt[3] + 2*x]) + 2*PolyLog[2, (I + Sqrt[
3] + (2*I)*x)/(2*Sqrt[3])])

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ({x}^{2}+x+1 \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x^2+x+1)^2,x)

[Out]

int(ln(x^2+x+1)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^2+x+1)^2,x, algorithm="maxima")

[Out]

x*log(x^2 + x + 1)^2 - integrate(2*(2*x^2 + x)*log(x^2 + x + 1)/(x^2 + x + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\log \left (x^{2} + x + 1\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^2+x+1)^2,x, algorithm="fricas")

[Out]

integral(log(x^2 + x + 1)^2, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RecursionError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x**2+x+1)**2,x)

[Out]

Exception raised: RecursionError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^2+x+1)^2,x, algorithm="giac")

[Out]

integrate(log(x^2 + x + 1)^2, x)