\(\int \frac {(-8-32 x+48 x^2) \log (x+2 x^2-2 x^3)}{-25 x-50 x^2+50 x^3} \, dx\) [3195]

Optimal result

Integrand size = 40, antiderivative size = 21 \[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=2+\frac {4}{25} \log ^2\left (x-2 x \left (-x+x^2\right )\right ) \]

[Out]

2+4/25*ln(x*(-2*x^2+2*x)+x)^2

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.68 (sec) , antiderivative size = 366, normalized size of antiderivative = 17.43, number of steps used = 42, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1608, 2608, 2604, 2404, 2338, 2353, 2352, 2354, 2438, 2465, 2441, 2437, 2440, 2439} \[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=-\frac {8}{25} \operatorname {PolyLog}\left (2,-\frac {-2 x-\sqrt {3}+1}{2 \sqrt {3}}\right )-\frac {8}{25} \operatorname {PolyLog}\left (2,\frac {-2 x+\sqrt {3}+1}{2 \sqrt {3}}\right )+\frac {8}{25} \log (x) \log \left (-2 x^3+2 x^2+x\right )+\frac {8}{25} \log \left (4 x-2 \left (1-\sqrt {3}\right )\right ) \log \left (-2 x^3+2 x^2+x\right )+\frac {8}{25} \log \left (4 x-2 \left (1+\sqrt {3}\right )\right ) \log \left (-2 x^3+2 x^2+x\right )-\frac {4}{25} \log ^2\left (-2 \left (-2 x-\sqrt {3}+1\right )\right )-\frac {4}{25} \log ^2\left (-2 \left (-2 x+\sqrt {3}+1\right )\right )-\frac {4 \log ^2(x)}{25}-\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )\right ) \log (x)-\frac {8}{25} \log \left (\frac {-2 x+\sqrt {3}+1}{2 \sqrt {3}}\right ) \log \left (4 x-2 \left (1-\sqrt {3}\right )\right )-\frac {8}{25} \log \left (-\frac {-2 x-\sqrt {3}+1}{2 \sqrt {3}}\right ) \log \left (4 x-2 \left (1+\sqrt {3}\right )\right )-\frac {8}{25} \log \left (\frac {2 x}{1+\sqrt {3}}\right ) \log \left (4 x-2 \left (1+\sqrt {3}\right )\right )-\frac {8}{25} \log \left (\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \log \left (4 x-2 \left (1+\sqrt {3}\right )\right )-\frac {8}{25} \log (x) \log \left (1-\frac {2 x}{1-\sqrt {3}}\right ) \]

[In]

Int[((-8 - 32*x + 48*x^2)*Log[x + 2*x^2 - 2*x^3])/(-25*x - 50*x^2 + 50*x^3),x]

[Out]

(-4*Log[-2*(1 - Sqrt[3] - 2*x)]^2)/25 - (4*Log[-2*(1 + Sqrt[3] - 2*x)]^2)/25 - (8*Log[-2*(1 - Sqrt[3])]*Log[x]
)/25 - (4*Log[x]^2)/25 - (8*Log[(1 + Sqrt[3] - 2*x)/(2*Sqrt[3])]*Log[-2*(1 - Sqrt[3]) + 4*x])/25 - (8*Log[(1 +
Sqrt[3])/2]*Log[-2*(1 + Sqrt[3]) + 4*x])/25 - (8*Log[-1/2*(1 - Sqrt[3] - 2*x)/Sqrt[3]]*Log[-2*(1 + Sqrt[3]) +
4*x])/25 - (8*Log[(2*x)/(1 + Sqrt[3])]*Log[-2*(1 + Sqrt[3]) + 4*x])/25 - (8*Log[x]*Log[1 - (2*x)/(1 - Sqrt[3]
)])/25 + (8*Log[x]*Log[x + 2*x^2 - 2*x^3])/25 + (8*Log[-2*(1 - Sqrt[3]) + 4*x]*Log[x + 2*x^2 - 2*x^3])/25 + (8
*Log[-2*(1 + Sqrt[3]) + 4*x]*Log[x + 2*x^2 - 2*x^3])/25 - (8*PolyLog[2, -1/2*(1 - Sqrt[3] - 2*x)/Sqrt[3]])/25
- (8*PolyLog[2, (1 + Sqrt[3] - 2*x)/(2*Sqrt[3])])/25

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 2439

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

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 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 \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{x \left (-25-50 x+50 x^2\right )} \, dx \\ & = \int \left (\frac {8 \log \left (x+2 x^2-2 x^3\right )}{25 x}+\frac {16 (-1+2 x) \log \left (x+2 x^2-2 x^3\right )}{25 \left (-1-2 x+2 x^2\right )}\right ) \, dx \\ & = \frac {8}{25} \int \frac {\log \left (x+2 x^2-2 x^3\right )}{x} \, dx+\frac {16}{25} \int \frac {(-1+2 x) \log \left (x+2 x^2-2 x^3\right )}{-1-2 x+2 x^2} \, dx \\ & = \frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \int \frac {\left (1+4 x-6 x^2\right ) \log (x)}{x+2 x^2-2 x^3} \, dx+\frac {16}{25} \int \left (\frac {2 \log \left (x+2 x^2-2 x^3\right )}{-2-2 \sqrt {3}+4 x}+\frac {2 \log \left (x+2 x^2-2 x^3\right )}{-2+2 \sqrt {3}+4 x}\right ) \, dx \\ & = \frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \int \frac {\left (1+4 x-6 x^2\right ) \log (x)}{x \left (1+2 x-2 x^2\right )} \, dx+\frac {32}{25} \int \frac {\log \left (x+2 x^2-2 x^3\right )}{-2-2 \sqrt {3}+4 x} \, dx+\frac {32}{25} \int \frac {\log \left (x+2 x^2-2 x^3\right )}{-2+2 \sqrt {3}+4 x} \, dx \\ & = \frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \int \left (\frac {\log (x)}{x}+\frac {2 (-1+2 x) \log (x)}{-1-2 x+2 x^2}\right ) \, dx-\frac {8}{25} \int \frac {\left (1+4 x-6 x^2\right ) \log \left (-2-2 \sqrt {3}+4 x\right )}{x+2 x^2-2 x^3} \, dx-\frac {8}{25} \int \frac {\left (1+4 x-6 x^2\right ) \log \left (-2+2 \sqrt {3}+4 x\right )}{x+2 x^2-2 x^3} \, dx \\ & = \frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \int \frac {\log (x)}{x} \, dx-\frac {8}{25} \int \frac {\left (1+4 x-6 x^2\right ) \log \left (-2-2 \sqrt {3}+4 x\right )}{x \left (1+2 x-2 x^2\right )} \, dx-\frac {8}{25} \int \frac {\left (1+4 x-6 x^2\right ) \log \left (-2+2 \sqrt {3}+4 x\right )}{x \left (1+2 x-2 x^2\right )} \, dx-\frac {16}{25} \int \frac {(-1+2 x) \log (x)}{-1-2 x+2 x^2} \, dx \\ & = -\frac {4}{25} \log ^2(x)+\frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \int \left (\frac {\log \left (-2-2 \sqrt {3}+4 x\right )}{x}+\frac {2 (-1+2 x) \log \left (-2-2 \sqrt {3}+4 x\right )}{-1-2 x+2 x^2}\right ) \, dx-\frac {8}{25} \int \left (\frac {\log \left (-2+2 \sqrt {3}+4 x\right )}{x}+\frac {2 (-1+2 x) \log \left (-2+2 \sqrt {3}+4 x\right )}{-1-2 x+2 x^2}\right ) \, dx-\frac {16}{25} \int \left (\frac {2 \log (x)}{-2-2 \sqrt {3}+4 x}+\frac {2 \log (x)}{-2+2 \sqrt {3}+4 x}\right ) \, dx \\ & = -\frac {4}{25} \log ^2(x)+\frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \int \frac {\log \left (-2-2 \sqrt {3}+4 x\right )}{x} \, dx-\frac {8}{25} \int \frac {\log \left (-2+2 \sqrt {3}+4 x\right )}{x} \, dx-\frac {16}{25} \int \frac {(-1+2 x) \log \left (-2-2 \sqrt {3}+4 x\right )}{-1-2 x+2 x^2} \, dx-\frac {16}{25} \int \frac {(-1+2 x) \log \left (-2+2 \sqrt {3}+4 x\right )}{-1-2 x+2 x^2} \, dx-\frac {32}{25} \int \frac {\log (x)}{-2-2 \sqrt {3}+4 x} \, dx-\frac {32}{25} \int \frac {\log (x)}{-2+2 \sqrt {3}+4 x} \, dx \\ & = -\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )\right ) \log (x)-\frac {4 \log ^2(x)}{25}-\frac {8}{25} \log \left (\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {2 x}{1+\sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log (x) \log \left (1-\frac {2 x}{1-\sqrt {3}}\right )+\frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {16}{25} \int \left (\frac {2 \log \left (-2-2 \sqrt {3}+4 x\right )}{-2-2 \sqrt {3}+4 x}+\frac {2 \log \left (-2-2 \sqrt {3}+4 x\right )}{-2+2 \sqrt {3}+4 x}\right ) \, dx-\frac {16}{25} \int \left (\frac {2 \log \left (-2+2 \sqrt {3}+4 x\right )}{-2-2 \sqrt {3}+4 x}+\frac {2 \log \left (-2+2 \sqrt {3}+4 x\right )}{-2+2 \sqrt {3}+4 x}\right ) \, dx-\frac {32}{25} \int \frac {\log \left (-\frac {4 x}{-2-2 \sqrt {3}}\right )}{-2-2 \sqrt {3}+4 x} \, dx+\frac {32}{25} \int \frac {\log \left (\frac {4 x}{2+2 \sqrt {3}}\right )}{-2-2 \sqrt {3}+4 x} \, dx \\ & = -\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )\right ) \log (x)-\frac {4 \log ^2(x)}{25}-\frac {8}{25} \log \left (\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {2 x}{1+\sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log (x) \log \left (1-\frac {2 x}{1-\sqrt {3}}\right )+\frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {32}{25} \int \frac {\log \left (-2-2 \sqrt {3}+4 x\right )}{-2-2 \sqrt {3}+4 x} \, dx-\frac {32}{25} \int \frac {\log \left (-2-2 \sqrt {3}+4 x\right )}{-2+2 \sqrt {3}+4 x} \, dx-\frac {32}{25} \int \frac {\log \left (-2+2 \sqrt {3}+4 x\right )}{-2-2 \sqrt {3}+4 x} \, dx-\frac {32}{25} \int \frac {\log \left (-2+2 \sqrt {3}+4 x\right )}{-2+2 \sqrt {3}+4 x} \, dx \\ & = -\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )\right ) \log (x)-\frac {4 \log ^2(x)}{25}-\frac {8}{25} \log \left (\frac {1+\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (-\frac {1-\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {2 x}{1+\sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log (x) \log \left (1-\frac {2 x}{1-\sqrt {3}}\right )+\frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2-2 \sqrt {3}+4 x\right )-\frac {8}{25} \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+2 \sqrt {3}+4 x\right )+\frac {32}{25} \int \frac {\log \left (\frac {4 \left (-2-2 \sqrt {3}+4 x\right )}{4 \left (-2-2 \sqrt {3}\right )-4 \left (-2+2 \sqrt {3}\right )}\right )}{-2+2 \sqrt {3}+4 x} \, dx+\frac {32}{25} \int \frac {\log \left (\frac {4 \left (-2+2 \sqrt {3}+4 x\right )}{-4 \left (-2-2 \sqrt {3}\right )+4 \left (-2+2 \sqrt {3}\right )}\right )}{-2-2 \sqrt {3}+4 x} \, dx \\ & = -\frac {4}{25} \log ^2\left (-2 \left (1-\sqrt {3}-2 x\right )\right )-\frac {4}{25} \log ^2\left (-2 \left (1+\sqrt {3}-2 x\right )\right )-\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )\right ) \log (x)-\frac {4 \log ^2(x)}{25}-\frac {8}{25} \log \left (\frac {1+\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (-\frac {1-\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {2 x}{1+\sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log (x) \log \left (1-\frac {2 x}{1-\sqrt {3}}\right )+\frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \text {Subst}\left (\int \frac {\log \left (1+\frac {4 x}{4 \left (-2-2 \sqrt {3}\right )-4 \left (-2+2 \sqrt {3}\right )}\right )}{x} \, dx,x,-2+2 \sqrt {3}+4 x\right )+\frac {8}{25} \text {Subst}\left (\int \frac {\log \left (1+\frac {4 x}{-4 \left (-2-2 \sqrt {3}\right )+4 \left (-2+2 \sqrt {3}\right )}\right )}{x} \, dx,x,-2-2 \sqrt {3}+4 x\right ) \\ & = -\frac {4}{25} \log ^2\left (-2 \left (1-\sqrt {3}-2 x\right )\right )-\frac {4}{25} \log ^2\left (-2 \left (1+\sqrt {3}-2 x\right )\right )-\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )\right ) \log (x)-\frac {4 \log ^2(x)}{25}-\frac {8}{25} \log \left (\frac {1+\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (-\frac {1-\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log \left (\frac {2 x}{1+\sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {8}{25} \log (x) \log \left (1-\frac {2 x}{1-\sqrt {3}}\right )+\frac {8}{25} \log (x) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\frac {8}{25} \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )-\frac {8}{25} \text {Li}_2\left (-\frac {1-\sqrt {3}-2 x}{2 \sqrt {3}}\right )-\frac {8}{25} \text {Li}_2\left (\frac {1+\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(293\) vs. \(2(21)=42\).

Time = 0.19 (sec) , antiderivative size = 293, normalized size of antiderivative = 13.95 \[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=\frac {8}{25} \left (-\log \left (-2 \left (1-\sqrt {3}\right )\right ) \log (x)-\log \left (\frac {1-\sqrt {3}-2 x}{1-\sqrt {3}}\right ) \log (x)-\frac {\log ^2(x)}{2}-\frac {1}{2} \log ^2\left (-2 \left (1-\sqrt {3}\right )+4 x\right )-\log \left (4 \sqrt {3}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\log \left (\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\log \left (-\frac {1-\sqrt {3}-2 x}{2 \sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\log \left (\frac {2 x}{1+\sqrt {3}}\right ) \log \left (-2 \left (1+\sqrt {3}\right )+4 x\right )-\frac {1}{2} \log ^2\left (-2 \left (1+\sqrt {3}\right )+4 x\right )+\log (x) \log \left (x+2 x^2-2 x^3\right )+\log \left (-2 \left (1-\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )+\log \left (-2 \left (1+\sqrt {3}\right )+4 x\right ) \log \left (x+2 x^2-2 x^3\right )\right ) \]

[In]

Integrate[((-8 - 32*x + 48*x^2)*Log[x + 2*x^2 - 2*x^3])/(-25*x - 50*x^2 + 50*x^3),x]

[Out]

(8*(-(Log[-2*(1 - Sqrt[3])]*Log[x]) - Log[(1 - Sqrt[3] - 2*x)/(1 - Sqrt[3])]*Log[x] - Log[x]^2/2 - Log[-2*(1 -
Sqrt[3]) + 4*x]^2/2 - Log[4*Sqrt[3]]*Log[-2*(1 + Sqrt[3]) + 4*x] - Log[(1 + Sqrt[3])/2]*Log[-2*(1 + Sqrt[3])
+ 4*x] - Log[-1/2*(1 - Sqrt[3] - 2*x)/Sqrt[3]]*Log[-2*(1 + Sqrt[3]) + 4*x] - Log[(2*x)/(1 + Sqrt[3])]*Log[-2*(
1 + Sqrt[3]) + 4*x] - Log[-2*(1 + Sqrt[3]) + 4*x]^2/2 + Log[x]*Log[x + 2*x^2 - 2*x^3] + Log[-2*(1 - Sqrt[3]) +
4*x]*Log[x + 2*x^2 - 2*x^3] + Log[-2*(1 + Sqrt[3]) + 4*x]*Log[x + 2*x^2 - 2*x^3]))/25

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {4 \ln \left (-2 x^{3}+2 x^{2}+x \right )^{2}}{25}\)	\(18\)
norman	\(\frac {4 \ln \left (-2 x^{3}+2 x^{2}+x \right )^{2}}{25}\)	\(18\)
risch	\(\frac {4 \ln \left (-2 x^{3}+2 x^{2}+x \right )^{2}}{25}\)	\(18\)
parts	\(\frac {8 \ln \left (-2 x^{3}+2 x^{2}+x \right ) \ln \left (x \left (2 x^{2}-2 x -1\right )\right )}{25}-\frac {4 {\ln \left (x \left (2 x^{2}-2 x -1\right )\right )}^{2}}{25}\)	\(47\)

[In]

int((48*x^2-32*x-8)*ln(-2*x^3+2*x^2+x)/(50*x^3-50*x^2-25*x),x,method=_RETURNVERBOSE)

[Out]

4/25*ln(-2*x^3+2*x^2+x)^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=\frac {4}{25} \, \log \left (-2 \, x^{3} + 2 \, x^{2} + x\right )^{2} \]

[In]

integrate((48*x^2-32*x-8)*log(-2*x^3+2*x^2+x)/(50*x^3-50*x^2-25*x),x, algorithm="fricas")

[Out]

4/25*log(-2*x^3 + 2*x^2 + x)^2

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=\frac {4 \log {\left (- 2 x^{3} + 2 x^{2} + x \right )}^{2}}{25} \]

[In]

integrate((48*x**2-32*x-8)*ln(-2*x**3+2*x**2+x)/(50*x**3-50*x**2-25*x),x)

[Out]

4*log(-2*x**3 + 2*x**2 + x)**2/25

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=\frac {4}{25} \, \log \left (-2 \, x^{2} + 2 \, x + 1\right )^{2} + \frac {8}{25} \, \log \left (-2 \, x^{2} + 2 \, x + 1\right ) \log \left (x\right ) + \frac {4}{25} \, \log \left (x\right )^{2} \]

[In]

integrate((48*x^2-32*x-8)*log(-2*x^3+2*x^2+x)/(50*x^3-50*x^2-25*x),x, algorithm="maxima")

[Out]

4/25*log(-2*x^2 + 2*x + 1)^2 + 8/25*log(-2*x^2 + 2*x + 1)*log(x) + 4/25*log(x)^2

Giac [F]

\[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=\int { \frac {8 \, {\left (6 \, x^{2} - 4 \, x - 1\right )} \log \left (-2 \, x^{3} + 2 \, x^{2} + x\right )}{25 \, {\left (2 \, x^{3} - 2 \, x^{2} - x\right )}} \,d x } \]

[In]

integrate((48*x^2-32*x-8)*log(-2*x^3+2*x^2+x)/(50*x^3-50*x^2-25*x),x, algorithm="giac")

[Out]

integrate(8/25*(6*x^2 - 4*x - 1)*log(-2*x^3 + 2*x^2 + x)/(2*x^3 - 2*x^2 - x), x)

Mupad [B] (verification not implemented)

Time = 9.46 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-8-32 x+48 x^2\right ) \log \left (x+2 x^2-2 x^3\right )}{-25 x-50 x^2+50 x^3} \, dx=\frac {4\,{\ln \left (x\,\left (-2\,x^2+2\,x+1\right )\right )}^2}{25} \]

[In]

int((log(x + 2*x^2 - 2*x^3)*(32*x - 48*x^2 + 8))/(25*x + 50*x^2 - 50*x^3),x)

[Out]

(4*log(x*(2*x - 2*x^2 + 1))^2)/25