\(\int \frac {x^2 \log (1+x+x^2)}{2+3 x+x^2} \, dx\) [98]

Optimal result

Integrand size = 21, antiderivative size = 311 \[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=-2 x+\sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{1+i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{3+i \sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{1-i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{3-i \sqrt {3}}\right )+\frac {1}{2} \log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+\log (2+2 x) \log \left (1+x+x^2\right )-4 \log (4+2 x) \log \left (1+x+x^2\right )-\operatorname {PolyLog}\left (2,\frac {2 (1+x)}{1-i \sqrt {3}}\right )-\operatorname {PolyLog}\left (2,\frac {2 (1+x)}{1+i \sqrt {3}}\right )+4 \operatorname {PolyLog}\left (2,\frac {2 (2+x)}{3-i \sqrt {3}}\right )+4 \operatorname {PolyLog}\left (2,\frac {2 (2+x)}{3+i \sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2608, 2603, 787, 648, 632, 210, 642, 2604, 2465, 2441, 2440, 2438} \[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=\sqrt {3} \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )-\operatorname {PolyLog}\left (2,\frac {2 (x+1)}{1-i \sqrt {3}}\right )-\operatorname {PolyLog}\left (2,\frac {2 (x+1)}{1+i \sqrt {3}}\right )+4 \operatorname {PolyLog}\left (2,\frac {2 (x+2)}{3-i \sqrt {3}}\right )+4 \operatorname {PolyLog}\left (2,\frac {2 (x+2)}{3+i \sqrt {3}}\right )+x \log \left (x^2+x+1\right )+\log (2 x+2) \log \left (x^2+x+1\right )-4 \log (2 x+4) \log \left (x^2+x+1\right )+\frac {1}{2} \log \left (x^2+x+1\right )-2 x-\log (2 x+2) \log \left (-\frac {2 x-i \sqrt {3}+1}{1+i \sqrt {3}}\right )+4 \log (2 x+4) \log \left (-\frac {2 x-i \sqrt {3}+1}{3+i \sqrt {3}}\right )-\log (2 x+2) \log \left (-\frac {2 x+i \sqrt {3}+1}{1-i \sqrt {3}}\right )+4 \log (2 x+4) \log \left (-\frac {2 x+i \sqrt {3}+1}{3-i \sqrt {3}}\right ) \]

[In]

Int[(x^2*Log[1 + x + x^2])/(2 + 3*x + x^2),x]

[Out]

-2*x + Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - Log[2 + 2*x]*Log[-((1 - I*Sqrt[3] + 2*x)/(1 + I*Sqrt[3]))] + 4*Log[
4 + 2*x]*Log[-((1 - I*Sqrt[3] + 2*x)/(3 + I*Sqrt[3]))] - Log[2 + 2*x]*Log[-((1 + I*Sqrt[3] + 2*x)/(1 - I*Sqrt[
3]))] + 4*Log[4 + 2*x]*Log[-((1 + I*Sqrt[3] + 2*x)/(3 - I*Sqrt[3]))] + Log[1 + x + x^2]/2 + x*Log[1 + x + x^2]
+ Log[2 + 2*x]*Log[1 + x + x^2] - 4*Log[4 + 2*x]*Log[1 + x + x^2] - PolyLog[2, (2*(1 + x))/(1 - I*Sqrt[3])] -
PolyLog[2, (2*(1 + x))/(1 + I*Sqrt[3])] + 4*PolyLog[2, (2*(2 + x))/(3 - I*Sqrt[3])] + 4*PolyLog[2, (2*(2 + x)
)/(3 + I*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 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*} \text {integral}& = \int \left (\log \left (1+x+x^2\right )-\frac {(2+3 x) \log \left (1+x+x^2\right )}{2+3 x+x^2}\right ) \, dx \\ & = \int \log \left (1+x+x^2\right ) \, dx-\int \frac {(2+3 x) \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx \\ & = x \log \left (1+x+x^2\right )-\int \frac {x (1+2 x)}{1+x+x^2} \, dx-\int \left (-\frac {2 \log \left (1+x+x^2\right )}{2+2 x}+\frac {8 \log \left (1+x+x^2\right )}{4+2 x}\right ) \, dx \\ & = -2 x+x \log \left (1+x+x^2\right )+2 \int \frac {\log \left (1+x+x^2\right )}{2+2 x} \, dx-8 \int \frac {\log \left (1+x+x^2\right )}{4+2 x} \, dx-\int \frac {-2-x}{1+x+x^2} \, dx \\ & = -2 x+x \log \left (1+x+x^2\right )+\log (2+2 x) \log \left (1+x+x^2\right )-4 \log (4+2 x) \log \left (1+x+x^2\right )+\frac {1}{2} \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {3}{2} \int \frac {1}{1+x+x^2} \, dx+4 \int \frac {(1+2 x) \log (4+2 x)}{1+x+x^2} \, dx-\int \frac {(1+2 x) \log (2+2 x)}{1+x+x^2} \, dx \\ & = -2 x+\frac {1}{2} \log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+\log (2+2 x) \log \left (1+x+x^2\right )-4 \log (4+2 x) \log \left (1+x+x^2\right )-3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+4 \int \left (\frac {2 \log (4+2 x)}{1-i \sqrt {3}+2 x}+\frac {2 \log (4+2 x)}{1+i \sqrt {3}+2 x}\right ) \, dx-\int \left (\frac {2 \log (2+2 x)}{1-i \sqrt {3}+2 x}+\frac {2 \log (2+2 x)}{1+i \sqrt {3}+2 x}\right ) \, dx \\ & = -2 x+\sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+\frac {1}{2} \log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+\log (2+2 x) \log \left (1+x+x^2\right )-4 \log (4+2 x) \log \left (1+x+x^2\right )-2 \int \frac {\log (2+2 x)}{1-i \sqrt {3}+2 x} \, dx-2 \int \frac {\log (2+2 x)}{1+i \sqrt {3}+2 x} \, dx+8 \int \frac {\log (4+2 x)}{1-i \sqrt {3}+2 x} \, dx+8 \int \frac {\log (4+2 x)}{1+i \sqrt {3}+2 x} \, dx \\ & = -2 x+\sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{1+i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{3+i \sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{1-i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{3-i \sqrt {3}}\right )+\frac {1}{2} \log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+\log (2+2 x) \log \left (1+x+x^2\right )-4 \log (4+2 x) \log \left (1+x+x^2\right )+2 \int \frac {\log \left (\frac {2 \left (1-i \sqrt {3}+2 x\right )}{-4+2 \left (1-i \sqrt {3}\right )}\right )}{2+2 x} \, dx+2 \int \frac {\log \left (\frac {2 \left (1+i \sqrt {3}+2 x\right )}{-4+2 \left (1+i \sqrt {3}\right )}\right )}{2+2 x} \, dx-8 \int \frac {\log \left (\frac {2 \left (1-i \sqrt {3}+2 x\right )}{-8+2 \left (1-i \sqrt {3}\right )}\right )}{4+2 x} \, dx-8 \int \frac {\log \left (\frac {2 \left (1+i \sqrt {3}+2 x\right )}{-8+2 \left (1+i \sqrt {3}\right )}\right )}{4+2 x} \, dx \\ & = -2 x+\sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{1+i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{3+i \sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{1-i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{3-i \sqrt {3}}\right )+\frac {1}{2} \log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+\log (2+2 x) \log \left (1+x+x^2\right )-4 \log (4+2 x) \log \left (1+x+x^2\right )-4 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-8+2 \left (1-i \sqrt {3}\right )}\right )}{x} \, dx,x,4+2 x\right )-4 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-8+2 \left (1+i \sqrt {3}\right )}\right )}{x} \, dx,x,4+2 x\right )+\text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-4+2 \left (1-i \sqrt {3}\right )}\right )}{x} \, dx,x,2+2 x\right )+\text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-4+2 \left (1+i \sqrt {3}\right )}\right )}{x} \, dx,x,2+2 x\right ) \\ & = -2 x+\sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{1+i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1-i \sqrt {3}+2 x}{3+i \sqrt {3}}\right )-\log (2+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{1-i \sqrt {3}}\right )+4 \log (4+2 x) \log \left (-\frac {1+i \sqrt {3}+2 x}{3-i \sqrt {3}}\right )+\frac {1}{2} \log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+\log (2+2 x) \log \left (1+x+x^2\right )-4 \log (4+2 x) \log \left (1+x+x^2\right )-\text {Li}_2\left (\frac {2 (1+x)}{1+i \sqrt {3}}\right )-\text {Li}_2\left (\frac {2 i (1+x)}{i+\sqrt {3}}\right )+4 \text {Li}_2\left (\frac {2 (2+x)}{3-i \sqrt {3}}\right )+4 \text {Li}_2\left (\frac {2 (2+x)}{3+i \sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=-2 x+\sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )-\log \left (\frac {-i+\sqrt {3}-2 i x}{i+\sqrt {3}}\right ) \log (2 (1+x))-\log \left (\frac {i+\sqrt {3}+2 i x}{-i+\sqrt {3}}\right ) \log (2 (1+x))+\frac {1}{2} \log \left (1+x+x^2\right )+x \log \left (1+x+x^2\right )+\log (2 (1+x)) \log \left (1+x+x^2\right )-4 \log (2 (2+x)) \log \left (1+x+x^2\right )-\operatorname {PolyLog}\left (2,\frac {2 (1+x)}{1+i \sqrt {3}}\right )-\operatorname {PolyLog}\left (2,\frac {2 i (1+x)}{i+\sqrt {3}}\right )+4 \left (\left (\log \left (\frac {-i+\sqrt {3}-2 i x}{3 i+\sqrt {3}}\right )+\log \left (\frac {i+\sqrt {3}+2 i x}{-3 i+\sqrt {3}}\right )\right ) \log (2 (2+x))+\operatorname {PolyLog}\left (2,\frac {2 (2+x)}{3+i \sqrt {3}}\right )+\operatorname {PolyLog}\left (2,\frac {2 i (2+x)}{3 i+\sqrt {3}}\right )\right ) \]

[In]

Integrate[(x^2*Log[1 + x + x^2])/(2 + 3*x + x^2),x]

[Out]

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

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.90

method	result	size
default	\(-2 x +\frac {\ln \left (x^{2}+x +1\right )}{2}+x \ln \left (x^{2}+x +1\right )+\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}-4 \ln \left (x +2\right ) \ln \left (x^{2}+x +1\right )+4 \ln \left (x +2\right ) \ln \left (\frac {-1-2 x +i \sqrt {3}}{3+i \sqrt {3}}\right )+4 \ln \left (x +2\right ) \ln \left (\frac {1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}\right )+4 \operatorname {dilog}\left (\frac {-1-2 x +i \sqrt {3}}{3+i \sqrt {3}}\right )+4 \operatorname {dilog}\left (\frac {1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}\right )+\ln \left (x +1\right ) \ln \left (x^{2}+x +1\right )-\ln \left (x +1\right ) \ln \left (\frac {-1-2 x +i \sqrt {3}}{1+i \sqrt {3}}\right )-\ln \left (x +1\right ) \ln \left (\frac {1+2 x +i \sqrt {3}}{i \sqrt {3}-1}\right )-\operatorname {dilog}\left (\frac {-1-2 x +i \sqrt {3}}{1+i \sqrt {3}}\right )-\operatorname {dilog}\left (\frac {1+2 x +i \sqrt {3}}{i \sqrt {3}-1}\right )\)	\(279\)
risch	\(-2 x +\frac {\ln \left (x^{2}+x +1\right )}{2}+x \ln \left (x^{2}+x +1\right )+\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}-4 \ln \left (x +2\right ) \ln \left (x^{2}+x +1\right )+4 \ln \left (x +2\right ) \ln \left (\frac {-1-2 x +i \sqrt {3}}{3+i \sqrt {3}}\right )+4 \ln \left (x +2\right ) \ln \left (\frac {1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}\right )+4 \operatorname {dilog}\left (\frac {-1-2 x +i \sqrt {3}}{3+i \sqrt {3}}\right )+4 \operatorname {dilog}\left (\frac {1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}\right )+\ln \left (x +1\right ) \ln \left (x^{2}+x +1\right )-\ln \left (x +1\right ) \ln \left (\frac {-1-2 x +i \sqrt {3}}{1+i \sqrt {3}}\right )-\ln \left (x +1\right ) \ln \left (\frac {1+2 x +i \sqrt {3}}{i \sqrt {3}-1}\right )-\operatorname {dilog}\left (\frac {-1-2 x +i \sqrt {3}}{1+i \sqrt {3}}\right )-\operatorname {dilog}\left (\frac {1+2 x +i \sqrt {3}}{i \sqrt {3}-1}\right )\)	\(279\)
parts	\(-2 x +\frac {\ln \left (x^{2}+x +1\right )}{2}+x \ln \left (x^{2}+x +1\right )+\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}-4 \ln \left (x +2\right ) \ln \left (x^{2}+x +1\right )+4 \ln \left (x +2\right ) \ln \left (\frac {-1-2 x +i \sqrt {3}}{3+i \sqrt {3}}\right )+4 \ln \left (x +2\right ) \ln \left (\frac {1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}\right )+4 \operatorname {dilog}\left (\frac {-1-2 x +i \sqrt {3}}{3+i \sqrt {3}}\right )+4 \operatorname {dilog}\left (\frac {1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}\right )+\ln \left (x +1\right ) \ln \left (x^{2}+x +1\right )-\ln \left (x +1\right ) \ln \left (\frac {-1-2 x +i \sqrt {3}}{1+i \sqrt {3}}\right )-\ln \left (x +1\right ) \ln \left (\frac {1+2 x +i \sqrt {3}}{i \sqrt {3}-1}\right )-\operatorname {dilog}\left (\frac {-1-2 x +i \sqrt {3}}{1+i \sqrt {3}}\right )-\operatorname {dilog}\left (\frac {1+2 x +i \sqrt {3}}{i \sqrt {3}-1}\right )\)	\(279\)

[In]

int(x^2*ln(x^2+x+1)/(x^2+3*x+2),x,method=_RETURNVERBOSE)

[Out]

-2*x+1/2*ln(x^2+x+1)+x*ln(x^2+x+1)+arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-4*ln(x+2)*ln(x^2+x+1)+4*ln(x+2)*ln((-1-
2*x+I*3^(1/2))/(3+I*3^(1/2)))+4*ln(x+2)*ln((1+2*x+I*3^(1/2))/(-3+I*3^(1/2)))+4*dilog((-1-2*x+I*3^(1/2))/(3+I*3
^(1/2)))+4*dilog((1+2*x+I*3^(1/2))/(-3+I*3^(1/2)))+ln(x+1)*ln(x^2+x+1)-ln(x+1)*ln((-1-2*x+I*3^(1/2))/(1+I*3^(1
/2)))-ln(x+1)*ln((1+2*x+I*3^(1/2))/(I*3^(1/2)-1))-dilog((-1-2*x+I*3^(1/2))/(1+I*3^(1/2)))-dilog((1+2*x+I*3^(1/
2))/(I*3^(1/2)-1))

Fricas [F]

\[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=\int { \frac {x^{2} \log \left (x^{2} + x + 1\right )}{x^{2} + 3 \, x + 2} \,d x } \]

[In]

integrate(x^2*log(x^2+x+1)/(x^2+3*x+2),x, algorithm="fricas")

[Out]

integral(x^2*log(x^2 + x + 1)/(x^2 + 3*x + 2), x)

Sympy [F]

\[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=\int \frac {x^{2} \log {\left (x^{2} + x + 1 \right )}}{\left (x + 1\right ) \left (x + 2\right )}\, dx \]

[In]

integrate(x**2*ln(x**2+x+1)/(x**2+3*x+2),x)

[Out]

Integral(x**2*log(x**2 + x + 1)/((x + 1)*(x + 2)), x)

Maxima [F]

\[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=\int { \frac {x^{2} \log \left (x^{2} + x + 1\right )}{x^{2} + 3 \, x + 2} \,d x } \]

[In]

integrate(x^2*log(x^2+x+1)/(x^2+3*x+2),x, algorithm="maxima")

[Out]

integrate(x^2*log(x^2 + x + 1)/(x^2 + 3*x + 2), x)

Giac [F]

\[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=\int { \frac {x^{2} \log \left (x^{2} + x + 1\right )}{x^{2} + 3 \, x + 2} \,d x } \]

[In]

integrate(x^2*log(x^2+x+1)/(x^2+3*x+2),x, algorithm="giac")

[Out]

integrate(x^2*log(x^2 + x + 1)/(x^2 + 3*x + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \log \left (1+x+x^2\right )}{2+3 x+x^2} \, dx=\int \frac {x^2\,\ln \left (x^2+x+1\right )}{x^2+3\,x+2} \,d x \]

[In]

int((x^2*log(x + x^2 + 1))/(3*x + x^2 + 2),x)

[Out]

int((x^2*log(x + x^2 + 1))/(3*x + x^2 + 2), x)