3.1.99 \(\int \log ^2(1+x+x^2) \, dx\) [99]

Optimal. Leaf size=371 \[ 8 x-4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) \log ^2\left (1-i \sqrt {3}+2 x\right )-\left (1+i \sqrt {3}\right ) \log \left (\frac {i \left (1-i \sqrt {3}+2 x\right )}{2 \sqrt {3}}\right ) \log \left (1+i \sqrt {3}+2 x\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \log ^2\left (1+i \sqrt {3}+2 x\right )-\left (1-i \sqrt {3}\right ) \log \left (1-i \sqrt {3}+2 x\right ) \log \left (-\frac {i \left (1+i \sqrt {3}+2 x\right )}{2 \sqrt {3}}\right )-2 \log \left (1+x+x^2\right )-4 x \log \left (1+x+x^2\right )+\left (1-i \sqrt {3}\right ) \log \left (1-i \sqrt {3}+2 x\right ) \log \left (1+x+x^2\right )+\left (1+i \sqrt {3}\right ) \log \left (1+i \sqrt {3}+2 x\right ) \log \left (1+x+x^2\right )+x \log ^2\left (1+x+x^2\right )-\left (1+i \sqrt {3}\right ) \text {Li}_2\left (-\frac {i-\sqrt {3}+2 i x}{2 \sqrt {3}}\right )-\left (1-i \sqrt {3}\right ) \text {Li}_2\left (\frac {i+\sqrt {3}+2 i x}{2 \sqrt {3}}\right ) \]

[Out]

8*x-2*ln(x^2+x+1)-4*x*ln(x^2+x+1)+x*ln(x^2+x+1)^2+ln(x^2+x+1)*ln(1+2*x-I*3^(1/2))*(1-I*3^(1/2))-1/2*ln(1+2*x-I
*3^(1/2))^2*(1-I*3^(1/2))-ln(1+2*x-I*3^(1/2))*ln(-1/6*I*(1+2*x+I*3^(1/2))*3^(1/2))*(1-I*3^(1/2))-polylog(2,1/6
*(I+2*I*x+3^(1/2))*3^(1/2))*(1-I*3^(1/2))+ln(x^2+x+1)*ln(1+2*x+I*3^(1/2))*(1+I*3^(1/2))-1/2*ln(1+2*x+I*3^(1/2)
)^2*(1+I*3^(1/2))-ln(1+2*x+I*3^(1/2))*ln(1/6*I*(1+2*x-I*3^(1/2))*3^(1/2))*(1+I*3^(1/2))-polylog(2,1/6*(-I-2*I*
x+3^(1/2))*3^(1/2))*(1+I*3^(1/2))-4*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 371, normalized size of antiderivative = 1.00, 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 = {2603, 2608, 787, 648, 632, 210, 642, 2604, 2465, 2437, 2338, 2441, 2440, 2438} \begin {gather*} -\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 )-4 \sqrt {3} \text {ArcTan}\left (\frac {2 x+1}{\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 ) \end {gather*}

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

Rule 210

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

Rule 632

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 642

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 648

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 787

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 2338

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 2437

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 2438

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]

Rule 2440

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 2441

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 2465

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 2603

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 2604

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 2608

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]

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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.11, size = 323, normalized size = 0.87 \begin {gather*} 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 )-\frac {1}{2} i \left (-i+\sqrt {3}\right ) \left (\log \left (1+i \sqrt {3}+2 x\right ) \left (2 \log \left (\frac {i+\sqrt {3}+2 i x}{2 \sqrt {3}}\right )+\log \left (1+i \sqrt {3}+2 x\right )\right )+2 \text {Li}_2\left (\frac {-i+\sqrt {3}-2 i x}{2 \sqrt {3}}\right )\right )+\frac {1}{2} i \left (i+\sqrt {3}\right ) \left (\log \left (1-i \sqrt {3}+2 x\right ) \left (2 \log \left (\frac {-i+\sqrt {3}-2 i x}{2 \sqrt {3}}\right )+\log \left (1-i \sqrt {3}+2 x\right )\right )+2 \text {Li}_2\left (\frac {i+\sqrt {3}+2 i x}{2 \sqrt {3}}\right )\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \[\int \ln \left (x^{2}+x +1\right )^{2}\, dx\]

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 = 0.00, 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(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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RecursionError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RecursionError >> maximum recursion depth exceeded

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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(log(x^2+x+1)^2,x, algorithm="giac")

[Out]

integrate(log(x^2 + x + 1)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\ln \left (x^2+x+1\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x + x^2 + 1)^2,x)

[Out]

int(log(x + x^2 + 1)^2, x)

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