\(\int \frac {\log ^2(-1+x+x^2)}{x^3} \, dx\) [100]

Optimal result

Integrand size = 13, antiderivative size = 443 \[ \int \frac {\log ^2\left (-1+x+x^2\right )}{x^3} \, dx=\log (x)-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right )+3 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{4} \left (3+\sqrt {5}\right ) \log ^2\left (1-\sqrt {5}+2 x\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 x}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{4} \left (3-\sqrt {5}\right ) \log ^2\left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )+3 \log (x) \log \left (1+\frac {2 x}{1+\sqrt {5}}\right )+\frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+3 \operatorname {PolyLog}\left (2,-\frac {2 x}{1+\sqrt {5}}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \operatorname {PolyLog}\left (2,-\frac {1-\sqrt {5}+2 x}{2 \sqrt {5}}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \operatorname {PolyLog}\left (2,\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )-3 \operatorname {PolyLog}\left (2,1+\frac {2 x}{1-\sqrt {5}}\right ) \]

[Out]

ln(x)+ln(x^2+x-1)/x-3*ln(x)*ln(x^2+x-1)-1/2*ln(x^2+x-1)^2/x^2+3*ln(1+2*x-5^(1/2))*ln(1/2*5^(1/2)-1/2)+3*ln(x)*
ln(1+2*x/(5^(1/2)+1))-3*polylog(2,1+2*x/(-5^(1/2)+1))+3*polylog(2,-2*x/(5^(1/2)+1))-1/2*ln(1+2*x+5^(1/2))*(-5^
(1/2)+1)+1/2*ln(x^2+x-1)*ln(1+2*x+5^(1/2))*(3-5^(1/2))-1/2*ln(1/10*(-1-2*x+5^(1/2))*5^(1/2))*ln(1+2*x+5^(1/2))
*(3-5^(1/2))-1/4*ln(1+2*x+5^(1/2))^2*(3-5^(1/2))-1/2*polylog(2,1/10*(1+2*x+5^(1/2))*5^(1/2))*(3-5^(1/2))-1/2*l
n(1+2*x-5^(1/2))*(5^(1/2)+1)+1/2*ln(x^2+x-1)*ln(1+2*x-5^(1/2))*(3+5^(1/2))-1/4*ln(1+2*x-5^(1/2))^2*(3+5^(1/2))
-1/2*ln(1+2*x-5^(1/2))*ln(1/10*(1+2*x+5^(1/2))*5^(1/2))*(3+5^(1/2))-1/2*polylog(2,1/10*(-1-2*x+5^(1/2))*5^(1/2
))*(3+5^(1/2))

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.231, Rules used = {2605, 2608, 814, 646, 31, 2604, 2404, 2353, 2352, 2354, 2438, 2465, 2437, 2338, 2441, 2440} \[ \int \frac {\log ^2\left (-1+x+x^2\right )}{x^3} \, dx=3 \operatorname {PolyLog}\left (2,-\frac {2 x}{1+\sqrt {5}}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \operatorname {PolyLog}\left (2,-\frac {2 x-\sqrt {5}+1}{2 \sqrt {5}}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \operatorname {PolyLog}\left (2,\frac {2 x+\sqrt {5}+1}{2 \sqrt {5}}\right )-3 \operatorname {PolyLog}\left (2,\frac {2 x}{1-\sqrt {5}}+1\right )-\frac {\log ^2\left (x^2+x-1\right )}{2 x^2}+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (x^2+x-1\right ) \log \left (2 x-\sqrt {5}+1\right )-3 \log (x) \log \left (x^2+x-1\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (2 x+\sqrt {5}+1\right ) \log \left (x^2+x-1\right )+\frac {\log \left (x^2+x-1\right )}{x}-\frac {1}{4} \left (3+\sqrt {5}\right ) \log ^2\left (2 x-\sqrt {5}+1\right )-\frac {1}{4} \left (3-\sqrt {5}\right ) \log ^2\left (2 x+\sqrt {5}+1\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (\frac {2 x+\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (2 x-\sqrt {5}+1\right )+3 \log \left (\frac {1}{2} \left (\sqrt {5}-1\right )\right ) \log \left (2 x-\sqrt {5}+1\right )-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (2 x-\sqrt {5}+1\right )+\log (x)-\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (-\frac {2 x-\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (2 x+\sqrt {5}+1\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (2 x+\sqrt {5}+1\right )+3 \log (x) \log \left (\frac {2 x}{1+\sqrt {5}}+1\right ) \]

[In]

Int[Log[-1 + x + x^2]^2/x^3,x]

[Out]

Log[x] - ((1 + Sqrt[5])*Log[1 - Sqrt[5] + 2*x])/2 + 3*Log[(-1 + Sqrt[5])/2]*Log[1 - Sqrt[5] + 2*x] - ((3 + Sqr
t[5])*Log[1 - Sqrt[5] + 2*x]^2)/4 - ((1 - Sqrt[5])*Log[1 + Sqrt[5] + 2*x])/2 - ((3 - Sqrt[5])*Log[-1/2*(1 - Sq
rt[5] + 2*x)/Sqrt[5]]*Log[1 + Sqrt[5] + 2*x])/2 - ((3 - Sqrt[5])*Log[1 + Sqrt[5] + 2*x]^2)/4 - ((3 + Sqrt[5])*
Log[1 - Sqrt[5] + 2*x]*Log[(1 + Sqrt[5] + 2*x)/(2*Sqrt[5])])/2 + 3*Log[x]*Log[1 + (2*x)/(1 + Sqrt[5])] + Log[-
1 + x + x^2]/x - 3*Log[x]*Log[-1 + x + x^2] + ((3 + Sqrt[5])*Log[1 - Sqrt[5] + 2*x]*Log[-1 + x + x^2])/2 + ((3
- Sqrt[5])*Log[1 + Sqrt[5] + 2*x]*Log[-1 + x + x^2])/2 - Log[-1 + x + x^2]^2/(2*x^2) + 3*PolyLog[2, (-2*x)/(1
+ Sqrt[5])] - ((3 + Sqrt[5])*PolyLog[2, -1/2*(1 - Sqrt[5] + 2*x)/Sqrt[5]])/2 - ((3 - Sqrt[5])*PolyLog[2, (1 +
Sqrt[5] + 2*x)/(2*Sqrt[5])])/2 - 3*PolyLog[2, 1 + (2*x)/(1 - Sqrt[5])]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 646

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), 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 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2353

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

Rule 2354

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

Rule 2404

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

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 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 2605

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

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}& = -\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+\int \frac {(1+2 x) \log \left (-1+x+x^2\right )}{x^2 \left (-1+x+x^2\right )} \, dx \\ & = -\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+\int \left (-\frac {\log \left (-1+x+x^2\right )}{x^2}-\frac {3 \log \left (-1+x+x^2\right )}{x}+\frac {(4+3 x) \log \left (-1+x+x^2\right )}{-1+x+x^2}\right ) \, dx \\ & = -\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}-3 \int \frac {\log \left (-1+x+x^2\right )}{x} \, dx-\int \frac {\log \left (-1+x+x^2\right )}{x^2} \, dx+\int \frac {(4+3 x) \log \left (-1+x+x^2\right )}{-1+x+x^2} \, dx \\ & = \frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+3 \int \frac {(1+2 x) \log (x)}{-1+x+x^2} \, dx-\int \frac {1+2 x}{x \left (-1+x+x^2\right )} \, dx+\int \left (\frac {\left (3+\sqrt {5}\right ) \log \left (-1+x+x^2\right )}{1-\sqrt {5}+2 x}+\frac {\left (3-\sqrt {5}\right ) \log \left (-1+x+x^2\right )}{1+\sqrt {5}+2 x}\right ) \, dx \\ & = \frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+3 \int \left (\frac {2 \log (x)}{1-\sqrt {5}+2 x}+\frac {2 \log (x)}{1+\sqrt {5}+2 x}\right ) \, dx+\left (3-\sqrt {5}\right ) \int \frac {\log \left (-1+x+x^2\right )}{1+\sqrt {5}+2 x} \, dx+\left (3+\sqrt {5}\right ) \int \frac {\log \left (-1+x+x^2\right )}{1-\sqrt {5}+2 x} \, dx-\int \left (-\frac {1}{x}+\frac {3+x}{-1+x+x^2}\right ) \, dx \\ & = \log (x)+\frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+6 \int \frac {\log (x)}{1-\sqrt {5}+2 x} \, dx+6 \int \frac {\log (x)}{1+\sqrt {5}+2 x} \, dx+\frac {1}{2} \left (-3-\sqrt {5}\right ) \int \frac {(1+2 x) \log \left (1-\sqrt {5}+2 x\right )}{-1+x+x^2} \, dx+\frac {1}{2} \left (-3+\sqrt {5}\right ) \int \frac {(1+2 x) \log \left (1+\sqrt {5}+2 x\right )}{-1+x+x^2} \, dx-\int \frac {3+x}{-1+x+x^2} \, dx \\ & = \log (x)+3 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 x\right )+3 \log (x) \log \left (1+\frac {2 x}{1+\sqrt {5}}\right )+\frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}-3 \int \frac {\log \left (1+\frac {2 x}{1+\sqrt {5}}\right )}{x} \, dx+6 \int \frac {\log \left (-\frac {2 x}{1-\sqrt {5}}\right )}{1-\sqrt {5}+2 x} \, dx+\frac {1}{2} \left (-3-\sqrt {5}\right ) \int \left (\frac {2 \log \left (1-\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (1-\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x}\right ) \, dx+\frac {1}{2} \left (-3+\sqrt {5}\right ) \int \left (\frac {2 \log \left (1+\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (1+\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x}\right ) \, dx+\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx-\frac {1}{2} \left (1+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx \\ & = \log (x)-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right )+3 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )+3 \log (x) \log \left (1+\frac {2 x}{1+\sqrt {5}}\right )+\frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+3 \text {Li}_2\left (-\frac {2 x}{1+\sqrt {5}}\right )-3 \text {Li}_2\left (1+\frac {2 x}{1-\sqrt {5}}\right )+\left (-3-\sqrt {5}\right ) \int \frac {\log \left (1-\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x} \, dx+\left (-3-\sqrt {5}\right ) \int \frac {\log \left (1-\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x} \, dx+\left (-3+\sqrt {5}\right ) \int \frac {\log \left (1+\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x} \, dx+\left (-3+\sqrt {5}\right ) \int \frac {\log \left (1+\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x} \, dx \\ & = \log (x)-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right )+3 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 x}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )+3 \log (x) \log \left (1+\frac {2 x}{1+\sqrt {5}}\right )+\frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+3 \text {Li}_2\left (-\frac {2 x}{1+\sqrt {5}}\right )-3 \text {Li}_2\left (1+\frac {2 x}{1-\sqrt {5}}\right )+\frac {1}{2} \left (-3-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-\sqrt {5}+2 x\right )+\left (3-\sqrt {5}\right ) \int \frac {\log \left (\frac {2 \left (1-\sqrt {5}+2 x\right )}{2 \left (1-\sqrt {5}\right )-2 \left (1+\sqrt {5}\right )}\right )}{1+\sqrt {5}+2 x} \, dx+\frac {1}{2} \left (-3+\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\sqrt {5}+2 x\right )+\left (3+\sqrt {5}\right ) \int \frac {\log \left (\frac {2 \left (1+\sqrt {5}+2 x\right )}{-2 \left (1-\sqrt {5}\right )+2 \left (1+\sqrt {5}\right )}\right )}{1-\sqrt {5}+2 x} \, dx \\ & = \log (x)-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right )+3 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{4} \left (3+\sqrt {5}\right ) \log ^2\left (1-\sqrt {5}+2 x\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 x}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{4} \left (3-\sqrt {5}\right ) \log ^2\left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )+3 \log (x) \log \left (1+\frac {2 x}{1+\sqrt {5}}\right )+\frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+3 \text {Li}_2\left (-\frac {2 x}{1+\sqrt {5}}\right )-3 \text {Li}_2\left (1+\frac {2 x}{1-\sqrt {5}}\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (1-\sqrt {5}\right )-2 \left (1+\sqrt {5}\right )}\right )}{x} \, dx,x,1+\sqrt {5}+2 x\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (1-\sqrt {5}\right )+2 \left (1+\sqrt {5}\right )}\right )}{x} \, dx,x,1-\sqrt {5}+2 x\right ) \\ & = \log (x)-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right )+3 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{4} \left (3+\sqrt {5}\right ) \log ^2\left (1-\sqrt {5}+2 x\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 x}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 x\right )-\frac {1}{4} \left (3-\sqrt {5}\right ) \log ^2\left (1+\sqrt {5}+2 x\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )+3 \log (x) \log \left (1+\frac {2 x}{1+\sqrt {5}}\right )+\frac {\log \left (-1+x+x^2\right )}{x}-3 \log (x) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )+\frac {1}{2} \left (3-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-\frac {\log ^2\left (-1+x+x^2\right )}{2 x^2}+3 \text {Li}_2\left (-\frac {2 x}{1+\sqrt {5}}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \text {Li}_2\left (-\frac {1-\sqrt {5}+2 x}{2 \sqrt {5}}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \text {Li}_2\left (\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )-3 \text {Li}_2\left (1+\frac {2 x}{1-\sqrt {5}}\right ) \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.86 \[ \int \frac {\log ^2\left (-1+x+x^2\right )}{x^3} \, dx=\frac {-2 \log ^2\left (-1+x+x^2\right )+x \left (4 x \log (x)-12 x \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log (x)-6 x \log \left (-1+\sqrt {5}-2 x\right ) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right )-2 \sqrt {5} x \log \left (-1+\sqrt {5}-2 x\right ) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right )+12 x \log (x) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right )-12 x \log \left (\frac {2 x}{-1+\sqrt {5}}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right )+3 x \log ^2\left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right )+\sqrt {5} x \log ^2\left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right )-6 x \log \left (-1+\sqrt {5}-2 x\right ) \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+x\right )-2 \sqrt {5} x \log \left (-1+\sqrt {5}-2 x\right ) \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+x\right )+12 x \log (x) \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+x\right )+3 x \log ^2\left (\frac {1}{2} \left (1+\sqrt {5}\right )+x\right )-\sqrt {5} x \log ^2\left (\frac {1}{2} \left (1+\sqrt {5}\right )+x\right )-2 x \log \left (1-\sqrt {5}+2 x\right )-2 \sqrt {5} x \log \left (1-\sqrt {5}+2 x\right )+3 x \log (5) \log \left (1-\sqrt {5}+2 x\right )+\sqrt {5} x \log (5) \log \left (1-\sqrt {5}+2 x\right )-2 x \log \left (1+\sqrt {5}+2 x\right )+2 \sqrt {5} x \log \left (1+\sqrt {5}+2 x\right )-6 x \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right ) \log \left (1+\sqrt {5}+2 x\right )+2 \sqrt {5} x \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right ) \log \left (1+\sqrt {5}+2 x\right )-6 x \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+x\right ) \log \left (1+\sqrt {5}+2 x\right )+2 \sqrt {5} x \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )+x\right ) \log \left (1+\sqrt {5}+2 x\right )+6 x \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right ) \log \left (\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )-2 \sqrt {5} x \log \left (\frac {1}{2}-\frac {\sqrt {5}}{2}+x\right ) \log \left (\frac {1+\sqrt {5}+2 x}{2 \sqrt {5}}\right )+4 \log \left (-1+x+x^2\right )+6 x \log \left (-1+\sqrt {5}-2 x\right ) \log \left (-1+x+x^2\right )+2 \sqrt {5} x \log \left (-1+\sqrt {5}-2 x\right ) \log \left (-1+x+x^2\right )-12 x \log (x) \log \left (-1+x+x^2\right )+6 x \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-2 \sqrt {5} x \log \left (1+\sqrt {5}+2 x\right ) \log \left (-1+x+x^2\right )-4 \sqrt {5} x \operatorname {PolyLog}\left (2,\frac {-1+\sqrt {5}-2 x}{2 \sqrt {5}}\right )-12 x \operatorname {PolyLog}\left (2,\frac {-1+\sqrt {5}-2 x}{-1+\sqrt {5}}\right )+12 x \operatorname {PolyLog}\left (2,-\frac {2 x}{1+\sqrt {5}}\right )\right )}{4 x^2} \]

[In]

Integrate[Log[-1 + x + x^2]^2/x^3,x]

[Out]

(-2*Log[-1 + x + x^2]^2 + x*(4*x*Log[x] - 12*x*Log[(1 + Sqrt[5])/2]*Log[x] - 6*x*Log[-1 + Sqrt[5] - 2*x]*Log[1
/2 - Sqrt[5]/2 + x] - 2*Sqrt[5]*x*Log[-1 + Sqrt[5] - 2*x]*Log[1/2 - Sqrt[5]/2 + x] + 12*x*Log[x]*Log[1/2 - Sqr
t[5]/2 + x] - 12*x*Log[(2*x)/(-1 + Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + x] + 3*x*Log[1/2 - Sqrt[5]/2 + x]^2 + Sqrt[
5]*x*Log[1/2 - Sqrt[5]/2 + x]^2 - 6*x*Log[-1 + Sqrt[5] - 2*x]*Log[(1 + Sqrt[5])/2 + x] - 2*Sqrt[5]*x*Log[-1 +
Sqrt[5] - 2*x]*Log[(1 + Sqrt[5])/2 + x] + 12*x*Log[x]*Log[(1 + Sqrt[5])/2 + x] + 3*x*Log[(1 + Sqrt[5])/2 + x]^
2 - Sqrt[5]*x*Log[(1 + Sqrt[5])/2 + x]^2 - 2*x*Log[1 - Sqrt[5] + 2*x] - 2*Sqrt[5]*x*Log[1 - Sqrt[5] + 2*x] + 3
*x*Log[5]*Log[1 - Sqrt[5] + 2*x] + Sqrt[5]*x*Log[5]*Log[1 - Sqrt[5] + 2*x] - 2*x*Log[1 + Sqrt[5] + 2*x] + 2*Sq
rt[5]*x*Log[1 + Sqrt[5] + 2*x] - 6*x*Log[1/2 - Sqrt[5]/2 + x]*Log[1 + Sqrt[5] + 2*x] + 2*Sqrt[5]*x*Log[1/2 - S
qrt[5]/2 + x]*Log[1 + Sqrt[5] + 2*x] - 6*x*Log[(1 + Sqrt[5])/2 + x]*Log[1 + Sqrt[5] + 2*x] + 2*Sqrt[5]*x*Log[(
1 + Sqrt[5])/2 + x]*Log[1 + Sqrt[5] + 2*x] + 6*x*Log[1/2 - Sqrt[5]/2 + x]*Log[(1 + Sqrt[5] + 2*x)/(2*Sqrt[5])]
- 2*Sqrt[5]*x*Log[1/2 - Sqrt[5]/2 + x]*Log[(1 + Sqrt[5] + 2*x)/(2*Sqrt[5])] + 4*Log[-1 + x + x^2] + 6*x*Log[-
1 + Sqrt[5] - 2*x]*Log[-1 + x + x^2] + 2*Sqrt[5]*x*Log[-1 + Sqrt[5] - 2*x]*Log[-1 + x + x^2] - 12*x*Log[x]*Log
[-1 + x + x^2] + 6*x*Log[1 + Sqrt[5] + 2*x]*Log[-1 + x + x^2] - 2*Sqrt[5]*x*Log[1 + Sqrt[5] + 2*x]*Log[-1 + x
+ x^2] - 4*Sqrt[5]*x*PolyLog[2, (-1 + Sqrt[5] - 2*x)/(2*Sqrt[5])] - 12*x*PolyLog[2, (-1 + Sqrt[5] - 2*x)/(-1 +
Sqrt[5])] + 12*x*PolyLog[2, (-2*x)/(1 + Sqrt[5])]))/(4*x^2)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.32 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.49

method	result	size
parts	\(-\frac {\ln \left (x^{2}+x -1\right )^{2}}{2 x^{2}}-3 \ln \left (x \right ) \ln \left (x^{2}+x -1\right )+3 \ln \left (x \right ) \ln \left (\frac {-1-2 x +\sqrt {5}}{\sqrt {5}-1}\right )+3 \ln \left (x \right ) \ln \left (\frac {1+2 x +\sqrt {5}}{\sqrt {5}+1}\right )+3 \operatorname {dilog}\left (\frac {-1-2 x +\sqrt {5}}{\sqrt {5}-1}\right )+3 \operatorname {dilog}\left (\frac {1+2 x +\sqrt {5}}{\sqrt {5}+1}\right )+\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (x^{2}+x -1\right )-\operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha +x +1}{2 \underline {\hspace {1.25 ex}}\alpha +1}\right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha +x +1}{2 \underline {\hspace {1.25 ex}}\alpha +1}\right )-\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{2}\right ) \left (\underline {\hspace {1.25 ex}}\alpha +2\right )\right )+\frac {\ln \left (x^{2}+x -1\right )}{x}+\ln \left (x \right )-\frac {\ln \left (x^{2}+x -1\right )}{2}+\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (1+2 x \right ) \sqrt {5}}{5}\right )\)	\(219\)

[In]

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

[Out]

-1/2*ln(x^2+x-1)^2/x^2-3*ln(x)*ln(x^2+x-1)+3*ln(x)*ln((-1-2*x+5^(1/2))/(5^(1/2)-1))+3*ln(x)*ln((1+2*x+5^(1/2))
/(5^(1/2)+1))+3*dilog((-1-2*x+5^(1/2))/(5^(1/2)-1))+3*dilog((1+2*x+5^(1/2))/(5^(1/2)+1))+Sum((ln(x-_alpha)*ln(
x^2+x-1)-dilog((_alpha+x+1)/(2*_alpha+1))-ln(x-_alpha)*ln((_alpha+x+1)/(2*_alpha+1))-1/2*ln(x-_alpha)^2)*(_alp
ha+2),_alpha=RootOf(_Z^2+_Z-1))+ln(x^2+x-1)/x+ln(x)-1/2*ln(x^2+x-1)+5^(1/2)*arctanh(1/5*(1+2*x)*5^(1/2))

Fricas [F]

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

[In]

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

[Out]

integral(log(x^2 + x - 1)^2/x^3, x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\log ^2\left (-1+x+x^2\right )}{x^3} \, dx=\text {Exception raised: RecursionError} \]

[In]

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

[Out]

Exception raised: RecursionError >> maximum recursion depth exceeded while calling a Python object

Maxima [F]

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

[In]

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

[Out]

-1/2*log(x^2 + x - 1)^2/x^2 + integrate((2*x + 1)*log(x^2 + x - 1)/(x^4 + x^3 - x^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(log(x^2 + x - 1)^2/x^3, x)

Mupad [F(-1)]

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

[In]

int(log(x + x^2 - 1)^2/x^3,x)

[Out]

int(log(x + x^2 - 1)^2/x^3, x)