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

Optimal. Leaf size=31 \[ \frac {x^2 \left (\frac {3 x}{2}-\log \left (\frac {x^3}{-1+x}\right )\right )^2}{2 \log ^2(5)} \]

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Rubi [A]  time = 0.46, antiderivative size = 62, normalized size of antiderivative = 2.00, number of steps used = 42, number of rules used = 16, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.213, Rules used = {12, 6742, 893, 2514, 2487, 29, 8, 2494, 2390, 2301, 2394, 2315, 2495, 43, 30, 2498} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[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 43

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 893

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 2301

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 2315

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

Rule 2390

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 2394

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 2487

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 2494

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 2495

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 2498

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 2514

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {\operatorname {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 {\operatorname {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 {aligned} \end {gather*}

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Mathematica [C]  time = 0.12, size = 142, normalized size = 4.58 \begin {gather*} \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 \text {Li}_2(1-x)+6 \text {Li}_2(x)}{2 \log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.52, size = 44, normalized size = 1.42 \begin {gather*} \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 \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-2*x)*log(x^3/(x-1))^2+(-9*x^3+13*x^2-6*x)*log(x^3/(x-1))+9*x^4-15*x^3+9*x^2)/(2*x-2)/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

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giac [A]  time = 0.32, size = 44, normalized size = 1.42 \begin {gather*} \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 \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-2*x)*log(x^3/(x-1))^2+(-9*x^3+13*x^2-6*x)*log(x^3/(x-1))+9*x^4-15*x^3+9*x^2)/(2*x-2)/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

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maple [A]  time = 0.52, size = 51, normalized size = 1.65

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.88, size = 63, normalized size = 2.03 \begin {gather*} \frac {9 \, x^{4} + 4 \, x^{2} \log \left (x - 1\right )^{2} - 36 \, x^{3} \log \relax (x) + 36 \, x^{2} \log \relax (x)^{2} + 12 \, {\left (x^{3} - 2 \, x^{2} \log \relax (x) - 1\right )} \log \left (x - 1\right ) + 12 \, \log \left (x - 1\right )}{8 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-2*x)*log(x^3/(x-1))^2+(-9*x^3+13*x^2-6*x)*log(x^3/(x-1))+9*x^4-15*x^3+9*x^2)/(2*x-2)/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

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mupad [B]  time = 2.13, size = 27, normalized size = 0.87 \begin {gather*} \frac {x^2\,{\left (3\,x-2\,\ln \left (\frac {x^3}{x-1}\right )\right )}^2}{8\,{\ln \relax (5)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.26, size = 51, normalized size = 1.65 \begin {gather*} \frac {9 x^{4}}{8 \log {\relax (5 )}^{2}} - \frac {3 x^{3} \log {\left (\frac {x^{3}}{x - 1} \right )}}{2 \log {\relax (5 )}^{2}} + \frac {x^{2} \log {\left (\frac {x^{3}}{x - 1} \right )}^{2}}{2 \log {\relax (5 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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