\(\int \frac {9 x^2-15 x^3+9 x^4+(-6 x+13 x^2-9 x^3) \log (\frac {x^3}{-1+x})+(-2 x+2 x^2) \log ^2(\frac {x^3}{-1+x})}{(-2+2 x) \log ^2(5)} \, dx\) [3239]

Optimal result

Integrand size = 75, antiderivative size = 31 \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {x^2 \left (\frac {3 x}{2}-\log \left (\frac {x^3}{-1+x}\right )\right )^2}{2 \log ^2(5)} \]

[Out]

1/2*x^2/ln(5)^2*(3/2*x-ln(x^3/(-1+x)))^2

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00, number of steps used = 42, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.213, Rules used = {12, 6874, 907, 2594, 2579, 29, 8, 2580, 2437, 2338, 2441, 2352, 2581, 45, 30, 2584} \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {9 x^4}{8 \log ^2(5)}-\frac {3 x^3 \log \left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {x^2 \log ^2\left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)} \]

[In]

Int[(9*x^2 - 15*x^3 + 9*x^4 + (-6*x + 13*x^2 - 9*x^3)*Log[x^3/(-1 + x)] + (-2*x + 2*x^2)*Log[x^3/(-1 + x)]^2)/
((-2 + 2*x)*Log[5]^2),x]

[Out]

(9*x^4)/(8*Log[5]^2) - (3*x^3*Log[-(x^3/(1 - x))])/(2*Log[5]^2) + (x^2*Log[-(x^3/(1 - x))]^2)/(2*Log[5]^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 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 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 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 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 2579

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[(a
+ b*x)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/b), x] + (Dist[q*r*s*((b*c - a*d)/b), Int[Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Dist[r*s*(p + q), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1
), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] &&
LtQ[s, 4]

Rule 2580

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]/((g_.) + (h_.)*(x_)), x_Sym
bol] :> Simp[Log[g + h*x]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/h), x] + (-Dist[b*p*(r/h), Int[Log[g + h*x]/(a
+ b*x), x], x] - Dist[d*q*(r/h), Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q,
r}, x] && NeQ[b*c - a*d, 0]

Rule 2581

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(m + 1))), x] + (-Dist[b*p*(r/(h
*(m + 1))), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[d*q*(r/(h*(m + 1))), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]

Rule 2584

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)*((g_.) + (h_.)*(x_))^(
m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(h*(m + 1))), x] + (-Dist[b*
p*r*(s/(h*(m + 1))), Int[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(a + b*x)), x], x] -
Dist[d*q*r*(s/(h*(m + 1))), Int[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x)), x]
, x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]

Rule 2594

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{-2+2 x} \, dx}{\log ^2(5)} \\ & = \frac {\int \left (\frac {3 x^2 \left (3-5 x+3 x^2\right )}{2 (-1+x)}-\frac {x \left (6-13 x+9 x^2\right ) \log \left (\frac {x^3}{-1+x}\right )}{2 (-1+x)}+x \log ^2\left (\frac {x^3}{-1+x}\right )\right ) \, dx}{\log ^2(5)} \\ & = -\frac {\int \frac {x \left (6-13 x+9 x^2\right ) \log \left (\frac {x^3}{-1+x}\right )}{-1+x} \, dx}{2 \log ^2(5)}+\frac {\int x \log ^2\left (\frac {x^3}{-1+x}\right ) \, dx}{\log ^2(5)}+\frac {3 \int \frac {x^2 \left (3-5 x+3 x^2\right )}{-1+x} \, dx}{2 \log ^2(5)} \\ & = \frac {x^2 \log ^2\left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}-\frac {\int \left (2 \log \left (\frac {x^3}{-1+x}\right )+\frac {2 \log \left (\frac {x^3}{-1+x}\right )}{-1+x}-4 x \log \left (\frac {x^3}{-1+x}\right )+9 x^2 \log \left (\frac {x^3}{-1+x}\right )\right ) \, dx}{2 \log ^2(5)}+\frac {\int \frac {x^2 \log \left (\frac {x^3}{-1+x}\right )}{-1+x} \, dx}{\log ^2(5)}+\frac {3 \int \left (1+\frac {1}{-1+x}+x-2 x^2+3 x^3\right ) \, dx}{2 \log ^2(5)}-\frac {3 \int x \log \left (\frac {x^3}{-1+x}\right ) \, dx}{\log ^2(5)} \\ & = \frac {3 x}{2 \log ^2(5)}+\frac {3 x^2}{4 \log ^2(5)}-\frac {x^3}{\log ^2(5)}+\frac {9 x^4}{8 \log ^2(5)}+\frac {3 \log (1-x)}{2 \log ^2(5)}-\frac {3 x^2 \log \left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {x^2 \log ^2\left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}-\frac {\int \log \left (\frac {x^3}{-1+x}\right ) \, dx}{\log ^2(5)}-\frac {\int \frac {\log \left (\frac {x^3}{-1+x}\right )}{-1+x} \, dx}{\log ^2(5)}+\frac {\int \left (\log \left (\frac {x^3}{-1+x}\right )+\frac {\log \left (\frac {x^3}{-1+x}\right )}{-1+x}+x \log \left (\frac {x^3}{-1+x}\right )\right ) \, dx}{\log ^2(5)}-\frac {3 \int \frac {x^2}{-1+x} \, dx}{2 \log ^2(5)}+\frac {2 \int x \log \left (\frac {x^3}{-1+x}\right ) \, dx}{\log ^2(5)}+\frac {9 \int x \, dx}{2 \log ^2(5)}-\frac {9 \int x^2 \log \left (\frac {x^3}{-1+x}\right ) \, dx}{2 \log ^2(5)} \\ & = \frac {3 x}{2 \log ^2(5)}+\frac {3 x^2}{\log ^2(5)}-\frac {x^3}{\log ^2(5)}+\frac {9 x^4}{8 \log ^2(5)}+\frac {3 \log (1-x)}{2 \log ^2(5)}+\frac {(1-x) \log \left (-\frac {x^3}{1-x}\right )}{\log ^2(5)}-\frac {x^2 \log \left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}-\frac {3 x^3 \log \left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}-\frac {\log (-1+x) \log \left (-\frac {x^3}{1-x}\right )}{\log ^2(5)}+\frac {x^2 \log ^2\left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {\int \frac {x^2}{-1+x} \, dx}{\log ^2(5)}-\frac {\int \frac {\log (-1+x)}{-1+x} \, dx}{\log ^2(5)}+\frac {\int \log \left (\frac {x^3}{-1+x}\right ) \, dx}{\log ^2(5)}+\frac {\int \frac {\log \left (\frac {x^3}{-1+x}\right )}{-1+x} \, dx}{\log ^2(5)}+\frac {\int x \log \left (\frac {x^3}{-1+x}\right ) \, dx}{\log ^2(5)}-\frac {3 \int \frac {x^3}{-1+x} \, dx}{2 \log ^2(5)}-\frac {3 \int \left (1+\frac {1}{-1+x}+x\right ) \, dx}{2 \log ^2(5)}+\frac {2 \int 1 \, dx}{\log ^2(5)}-\frac {3 \int \frac {1}{x} \, dx}{\log ^2(5)}-\frac {3 \int x \, dx}{\log ^2(5)}+\frac {3 \int \frac {\log (-1+x)}{x} \, dx}{\log ^2(5)}+\frac {9 \int x^2 \, dx}{2 \log ^2(5)} \\ & = \frac {2 x}{\log ^2(5)}+\frac {3 x^2}{4 \log ^2(5)}+\frac {x^3}{2 \log ^2(5)}+\frac {9 x^4}{8 \log ^2(5)}-\frac {3 \log (x)}{\log ^2(5)}+\frac {3 \log (-1+x) \log (x)}{\log ^2(5)}-\frac {3 x^3 \log \left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {x^2 \log ^2\left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {\int \frac {x^2}{-1+x} \, dx}{2 \log ^2(5)}+\frac {\int \left (1+\frac {1}{-1+x}+x\right ) \, dx}{\log ^2(5)}+\frac {\int \frac {\log (-1+x)}{-1+x} \, dx}{\log ^2(5)}-\frac {\text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-1+x\right )}{\log ^2(5)}-\frac {3 \int x \, dx}{2 \log ^2(5)}-\frac {3 \int \left (1+\frac {1}{-1+x}+x+x^2\right ) \, dx}{2 \log ^2(5)}-\frac {2 \int 1 \, dx}{\log ^2(5)}+\frac {3 \int \frac {1}{x} \, dx}{\log ^2(5)}-\frac {3 \int \frac {\log (-1+x)}{x} \, dx}{\log ^2(5)}-\frac {3 \int \frac {\log (x)}{-1+x} \, dx}{\log ^2(5)} \\ & = -\frac {x}{2 \log ^2(5)}-\frac {x^2}{4 \log ^2(5)}+\frac {9 x^4}{8 \log ^2(5)}-\frac {\log (1-x)}{2 \log ^2(5)}-\frac {\log ^2(-1+x)}{2 \log ^2(5)}-\frac {3 x^3 \log \left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {x^2 \log ^2\left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {3 \text {Li}_2(1-x)}{\log ^2(5)}+\frac {\int \left (1+\frac {1}{-1+x}+x\right ) \, dx}{2 \log ^2(5)}+\frac {\text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-1+x\right )}{\log ^2(5)}+\frac {3 \int \frac {\log (x)}{-1+x} \, dx}{\log ^2(5)} \\ & = \frac {9 x^4}{8 \log ^2(5)}-\frac {3 x^3 \log \left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)}+\frac {x^2 \log ^2\left (-\frac {x^3}{1-x}\right )}{2 \log ^2(5)} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 4.58 \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {\frac {9 x^4}{4}-3 \log (1-x)+\log ^2(1-x)+3 \log (-1+x)-\log ^2(-1+x)+6 \log (-1+x) \log (x)-3 x^3 \log \left (-\frac {x^3}{1-x}\right )+2 \log (1-x) \log \left (-\frac {x^3}{1-x}\right )+x^2 \log ^2\left (-\frac {x^3}{1-x}\right )-2 \log (-1+x) \log \left (\frac {x^3}{-1+x}\right )+6 \operatorname {PolyLog}(2,1-x)+6 \operatorname {PolyLog}(2,x)}{2 \log ^2(5)} \]

[In]

Integrate[(9*x^2 - 15*x^3 + 9*x^4 + (-6*x + 13*x^2 - 9*x^3)*Log[x^3/(-1 + x)] + (-2*x + 2*x^2)*Log[x^3/(-1 + x
)]^2)/((-2 + 2*x)*Log[5]^2),x]

[Out]

((9*x^4)/4 - 3*Log[1 - x] + Log[1 - x]^2 + 3*Log[-1 + x] - Log[-1 + x]^2 + 6*Log[-1 + x]*Log[x] - 3*x^3*Log[-(
x^3/(1 - x))] + 2*Log[1 - x]*Log[-(x^3/(1 - x))] + x^2*Log[-(x^3/(1 - x))]^2 - 2*Log[-1 + x]*Log[x^3/(-1 + x)]
+ 6*PolyLog[2, 1 - x] + 6*PolyLog[2, x])/(2*Log[5]^2)

Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42

method	result	size
parallelrisch	\(\frac {\frac {\ln \left (\frac {x^{3}}{-1+x}\right )^{2} x^{2}}{2}-\frac {3 \ln \left (\frac {x^{3}}{-1+x}\right ) x^{3}}{2}+\frac {9 x^{4}}{8}}{\ln \left (5\right )^{2}}\)	\(44\)
risch	\(\frac {\ln \left (\frac {x^{3}}{-1+x}\right )^{2} x^{2}}{2 \ln \left (5\right )^{2}}-\frac {3 \ln \left (\frac {x^{3}}{-1+x}\right ) x^{3}}{2 \ln \left (5\right )^{2}}+\frac {9 x^{4}}{8 \ln \left (5\right )^{2}}\)	\(51\)
norman	\(\frac {\frac {9 x^{4}}{8 \ln \left (5\right )}+\frac {x^{2} \ln \left (\frac {x^{3}}{-1+x}\right )^{2}}{2 \ln \left (5\right )}-\frac {3 x^{3} \ln \left (\frac {x^{3}}{-1+x}\right )}{2 \ln \left (5\right )}}{\ln \left (5\right )}\)	\(56\)

[In]

int(((2*x^2-2*x)*ln(x^3/(-1+x))^2+(-9*x^3+13*x^2-6*x)*ln(x^3/(-1+x))+9*x^4-15*x^3+9*x^2)/(-2+2*x)/ln(5)^2,x,me
thod=_RETURNVERBOSE)

[Out]

1/ln(5)^2*(1/2*ln(x^3/(-1+x))^2*x^2-3/2*ln(x^3/(-1+x))*x^3+9/8*x^4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {9 \, x^{4} - 12 \, x^{3} \log \left (\frac {x^{3}}{x - 1}\right ) + 4 \, x^{2} \log \left (\frac {x^{3}}{x - 1}\right )^{2}}{8 \, \log \left (5\right )^{2}} \]

[In]

integrate(((2*x^2-2*x)*log(x^3/(-1+x))^2+(-9*x^3+13*x^2-6*x)*log(x^3/(-1+x))+9*x^4-15*x^3+9*x^2)/(-2+2*x)/log(
5)^2,x, algorithm="fricas")

[Out]

1/8*(9*x^4 - 12*x^3*log(x^3/(x - 1)) + 4*x^2*log(x^3/(x - 1))^2)/log(5)^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {9 x^{4}}{8 \log {\left (5 \right )}^{2}} - \frac {3 x^{3} \log {\left (\frac {x^{3}}{x - 1} \right )}}{2 \log {\left (5 \right )}^{2}} + \frac {x^{2} \log {\left (\frac {x^{3}}{x - 1} \right )}^{2}}{2 \log {\left (5 \right )}^{2}} \]

[In]

integrate(((2*x**2-2*x)*ln(x**3/(-1+x))**2+(-9*x**3+13*x**2-6*x)*ln(x**3/(-1+x))+9*x**4-15*x**3+9*x**2)/(-2+2*
x)/ln(5)**2,x)

[Out]

9*x**4/(8*log(5)**2) - 3*x**3*log(x**3/(x - 1))/(2*log(5)**2) + x**2*log(x**3/(x - 1))**2/(2*log(5)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).

Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {9 \, x^{4} + 4 \, x^{2} \log \left (x - 1\right )^{2} - 36 \, x^{3} \log \left (x\right ) + 36 \, x^{2} \log \left (x\right )^{2} + 12 \, {\left (x^{3} - 2 \, x^{2} \log \left (x\right ) - 1\right )} \log \left (x - 1\right ) + 12 \, \log \left (x - 1\right )}{8 \, \log \left (5\right )^{2}} \]

[In]

integrate(((2*x^2-2*x)*log(x^3/(-1+x))^2+(-9*x^3+13*x^2-6*x)*log(x^3/(-1+x))+9*x^4-15*x^3+9*x^2)/(-2+2*x)/log(
5)^2,x, algorithm="maxima")

[Out]

1/8*(9*x^4 + 4*x^2*log(x - 1)^2 - 36*x^3*log(x) + 36*x^2*log(x)^2 + 12*(x^3 - 2*x^2*log(x) - 1)*log(x - 1) + 1
2*log(x - 1))/log(5)^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {9 \, x^{4} - 12 \, x^{3} \log \left (\frac {x^{3}}{x - 1}\right ) + 4 \, x^{2} \log \left (\frac {x^{3}}{x - 1}\right )^{2}}{8 \, \log \left (5\right )^{2}} \]

[In]

integrate(((2*x^2-2*x)*log(x^3/(-1+x))^2+(-9*x^3+13*x^2-6*x)*log(x^3/(-1+x))+9*x^4-15*x^3+9*x^2)/(-2+2*x)/log(
5)^2,x, algorithm="giac")

[Out]

1/8*(9*x^4 - 12*x^3*log(x^3/(x - 1)) + 4*x^2*log(x^3/(x - 1))^2)/log(5)^2

Mupad [B] (verification not implemented)

Time = 9.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {9 x^2-15 x^3+9 x^4+\left (-6 x+13 x^2-9 x^3\right ) \log \left (\frac {x^3}{-1+x}\right )+\left (-2 x+2 x^2\right ) \log ^2\left (\frac {x^3}{-1+x}\right )}{(-2+2 x) \log ^2(5)} \, dx=\frac {x^2\,{\left (3\,x-2\,\ln \left (\frac {x^3}{x-1}\right )\right )}^2}{8\,{\ln \left (5\right )}^2} \]

[In]

int(-(log(x^3/(x - 1))^2*(2*x - 2*x^2) - 9*x^2 + 15*x^3 - 9*x^4 + log(x^3/(x - 1))*(6*x - 13*x^2 + 9*x^3))/(lo
g(5)^2*(2*x - 2)),x)

[Out]

(x^2*(3*x - 2*log(x^3/(x - 1)))^2)/(8*log(5)^2)