3.89.63 $\int (2x^2-4x\log (2)+2{\log }^2(2)+(3x^2-4x\log (2)+{\log }^2(2))\log (\frac{x^2}{{\log }^4(3)})+(-12x+12\log (2))\log (x^3)\log (\frac{x^2}{{\log }^4(3)})+12{\log }^3(x^3)\log (\frac{x^2}{{\log }^4(3)})+{\log }^4(x^3)(2+\log (\frac{x^2}{{\log }^4(3)}))+{\log }^2(x^3)(-4x+4\log (2)+(-4x+2\log (2))\log (\frac{x^2}{{\log }^4(3)})))dx$

Optimal. Leaf size=25 $$x{(-x+\log (2)+{\log }^2\left(x^3\right))}^2\log \left(\frac{x^2}{{\log }^4(3)}\right)$$

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Rubi [B]  time = 0.62, antiderivative size = 368, normalized size of antiderivative = 14.72, number of steps used = 55, number of rules used = 13, integrand size = 132, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.098, Rules used = {2356, 2304, 2295, 2313, 2557, 12, 2296, 2361, 6742, 2330, 2305, 2366, 14} $$\begin{array}{c}-2x{\log }^4\left(x^3\right)+24x{\log }^3\left(x^3\right)-216x{\log }^2\left(x^3\right)+1296x\log \left(x^3\right)+24x\log (2)\log \left(x^3\right)-9x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+36x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(\log \left(\frac{x^2}{{\log }^4(3)}\right)+2)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x(\log \left(\frac{x^2}{{\log }^4(3)}\right)+2){\log }^4\left(x^3\right)+648x\log \left(\frac{x^2}{{\log }^4(3)}\right)\log \left(x^3\right)-648x(\log \left(\frac{x^2}{{\log }^4(3)}\right)+2)\log \left(x^3\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x\log \left(\frac{x^2}{{\log }^4(3)}\right){\log }^3\left(x^3\right)-12x(\log \left(\frac{x^2}{{\log }^4(3)}\right)+2){\log }^3\left(x^3\right)-2x^2\log \left(\frac{x^2}{{\log }^4(3)}\right){\log }^2\left(x^3\right)-108x\log \left(\frac{x^2}{{\log }^4(3)}\right){\log }^2\left(x^3\right)+2x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right){\log }^2\left(x^3\right)+108x(\log \left(\frac{x^2}{{\log }^4(3)}\right)+2){\log }^2\left(x^3\right)-6x^2\log \left(x^3\right)+6(x^2-2x\log (4))\log \left(x^3\right)-3888x+72x\log (4)-144x\log (2)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[2*x^2 - 4*x*Log[2] + 2*Log[2]^2 + (3*x^2 - 4*x*Log[2] + Log[2]^2)*Log[x^2/Log[3]^4] + (-12*x + 12*Log[2])*
Log[x^3]*Log[x^2/Log[3]^4] + 12*Log[x^3]^3*Log[x^2/Log[3]^4] + Log[x^3]^4*(2 + Log[x^2/Log[3]^4]) + Log[x^3]^2
*(-4*x + 4*Log[2] + (-4*x + 2*Log[2])*Log[x^2/Log[3]^4]),x]

[Out]

-3888*x - 144*x*Log[2] + 72*x*Log[4] + 1296*x*Log[x^3] - 6*x^2*Log[x^3] + 24*x*Log[2]*Log[x^3] + 6*(x^2 - 2*x*
Log[4])*Log[x^3] - 216*x*Log[x^3]^2 + 24*x*Log[x^3]^3 - 2*x*Log[x^3]^4 - 1944*x*Log[x^2/Log[3]^4] - 9*x^2*Log[
x^2/Log[3]^4] + x^3*Log[x^2/Log[3]^4] + 36*x*Log[2]*Log[x^2/Log[3]^4] - 2*x^2*Log[2]*Log[x^2/Log[3]^4] + x*Log
[2]^2*Log[x^2/Log[3]^4] + 9*(x^2 - 2*x*Log[4])*Log[x^2/Log[3]^4] + 648*x*Log[x^3]*Log[x^2/Log[3]^4] - 108*x*Lo
g[x^3]^2*Log[x^2/Log[3]^4] - 2*x^2*Log[x^3]^2*Log[x^2/Log[3]^4] + 2*x*Log[2]*Log[x^3]^2*Log[x^2/Log[3]^4] + 12
*x*Log[x^3]^3*Log[x^2/Log[3]^4] + 1944*x*(2 + Log[x^2/Log[3]^4]) - 648*x*Log[x^3]*(2 + Log[x^2/Log[3]^4]) + 10
8*x*Log[x^3]^2*(2 + Log[x^2/Log[3]^4]) - 12*x*Log[x^3]^3*(2 + Log[x^2/Log[3]^4]) + x*Log[x^3]^4*(2 + Log[x^2/L
og[3]^4])

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

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

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rule 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 6742

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{2x^3}{3}-2x^2\log (2)+2x{\log }^2(2)+12\int {\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)dx+\int (3x^2-4x\log (2)+{\log }^2(2))\log \left(\frac{x^2}{{\log }^4(3)}\right)dx+\int (-12x+12\log (2))\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)dx+\int {\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))dx+\int {\log }^2\left(x^3\right)(-4x+4\log (2)+(-4x+2\log (2))\log \left(\frac{x^2}{{\log }^4(3)}\right))dx\\ & =\frac{2x^3}{3}-2x^2\log (2)+2x{\log }^2(2)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-6x^2\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x\log (2)\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-2\int (1944-648\log \left(x^3\right)+108{\log }^2\left(x^3\right)-12{\log }^3\left(x^3\right)+{\log }^4\left(x^3\right))dx-24\int (-162+54\log \left(x^3\right)-9{\log }^2\left(x^3\right)+{\log }^3\left(x^3\right))dx-\int 12(-x+\log (4))\log \left(x^3\right)dx-\int 18(-x+\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)dx+\int (3x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)-4x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right))dx+\int (-4(x-\log (2)){\log }^2\left(x^3\right)-2(2x-\log (2)){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right))dx\\ & =\frac{2x^3}{3}-2x^2\log (2)+2x{\log }^2(2)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-6x^2\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x\log (2)\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-2\int {\log }^4\left(x^3\right)dx-2\int (2x-\log (2)){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)dx+3\int x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)dx-4\int (x-\log (2)){\log }^2\left(x^3\right)dx-12\int (-x+\log (4))\log \left(x^3\right)dx-18\int (-x+\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)dx-(4\log (2))\int x\log \left(\frac{x^2}{{\log }^4(3)}\right)dx+{\log }^2(2)\int \log \left(\frac{x^2}{{\log }^4(3)}\right)dx\\ & =6(x^2-2x\log (4))\log \left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-6x^2\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x\log (2)\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-2\int (2x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-\log (2){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right))dx-4\int (x{\log }^2\left(x^3\right)-\log (2){\log }^2\left(x^3\right))dx+24\int {\log }^3\left(x^3\right)dx+2(36\int (-\frac{x}{2}+\log (4))dx)\\ & =2(-9x^2+36x\log (4))+6(x^2-2x\log (4))\log \left(x^3\right)+24x{\log }^3\left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-6x^2\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x\log (2)\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-4\int x{\log }^2\left(x^3\right)dx-4\int x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)dx-216\int {\log }^2\left(x^3\right)dx+(2\log (2))\int {\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)dx+(4\log (2))\int {\log }^2\left(x^3\right)dx\\ & =2(-9x^2+36x\log (4))+6(x^2-2x\log (4))\log \left(x^3\right)-216x{\log }^2\left(x^3\right)-2x^2{\log }^2\left(x^3\right)+4x\log (2){\log }^2\left(x^3\right)+24x{\log }^3\left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)-9x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)+36x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+2x\log (2){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+8\int \frac{1}{4}x(9-6\log \left(x^3\right)+2{\log }^2\left(x^3\right))dx+12\int x\log \left(x^3\right)dx+1296\int \log \left(x^3\right)dx-(4\log (2))\int (18-6\log \left(x^3\right)+{\log }^2\left(x^3\right))dx-(24\log (2))\int \log \left(x^3\right)dx\\ & =-3888x-9x^2+2(-9x^2+36x\log (4))+1296x\log \left(x^3\right)+6x^2\log \left(x^3\right)-24x\log (2)\log \left(x^3\right)+6(x^2-2x\log (4))\log \left(x^3\right)-216x{\log }^2\left(x^3\right)-2x^2{\log }^2\left(x^3\right)+4x\log (2){\log }^2\left(x^3\right)+24x{\log }^3\left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)-9x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)+36x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+2x\log (2){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+2\int x(9-6\log \left(x^3\right)+2{\log }^2\left(x^3\right))dx-(4\log (2))\int {\log }^2\left(x^3\right)dx+(24\log (2))\int \log \left(x^3\right)dx\\ & =-3888x-9x^2-72x\log (2)+2(-9x^2+36x\log (4))+1296x\log \left(x^3\right)+6x^2\log \left(x^3\right)+6(x^2-2x\log (4))\log \left(x^3\right)-216x{\log }^2\left(x^3\right)-2x^2{\log }^2\left(x^3\right)+24x{\log }^3\left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)-9x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)+36x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+2x\log (2){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+2\int (9x-6x\log \left(x^3\right)+2x{\log }^2\left(x^3\right))dx+(24\log (2))\int \log \left(x^3\right)dx\\ & =-3888x-144x\log (2)+2(-9x^2+36x\log (4))+1296x\log \left(x^3\right)+6x^2\log \left(x^3\right)+24x\log (2)\log \left(x^3\right)+6(x^2-2x\log (4))\log \left(x^3\right)-216x{\log }^2\left(x^3\right)-2x^2{\log }^2\left(x^3\right)+24x{\log }^3\left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)-9x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)+36x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+2x\log (2){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+4\int x{\log }^2\left(x^3\right)dx-12\int x\log \left(x^3\right)dx\\ & =-3888x+9x^2-144x\log (2)+2(-9x^2+36x\log (4))+1296x\log \left(x^3\right)+24x\log (2)\log \left(x^3\right)+6(x^2-2x\log (4))\log \left(x^3\right)-216x{\log }^2\left(x^3\right)+24x{\log }^3\left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)-9x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)+36x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+2x\log (2){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12\int x\log \left(x^3\right)dx\\ & =-3888x+18x^2-144x\log (2)+2(-9x^2+36x\log (4))+1296x\log \left(x^3\right)-6x^2\log \left(x^3\right)+24x\log (2)\log \left(x^3\right)+6(x^2-2x\log (4))\log \left(x^3\right)-216x{\log }^2\left(x^3\right)+24x{\log }^3\left(x^3\right)-2x{\log }^4\left(x^3\right)-1944x\log \left(\frac{x^2}{{\log }^4(3)}\right)-9x^2\log \left(\frac{x^2}{{\log }^4(3)}\right)+x^3\log \left(\frac{x^2}{{\log }^4(3)}\right)+36x\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2\log (2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+x{\log }^2(2)\log \left(\frac{x^2}{{\log }^4(3)}\right)+9(x^2-2x\log (4))\log \left(\frac{x^2}{{\log }^4(3)}\right)+648x\log \left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-108x{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)-2x^2{\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+2x\log (2){\log }^2\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+12x{\log }^3\left(x^3\right)\log \left(\frac{x^2}{{\log }^4(3)}\right)+1944x(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-648x\log \left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+108x{\log }^2\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))-12x{\log }^3\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))+x{\log }^4\left(x^3\right)(2+\log \left(\frac{x^2}{{\log }^4(3)}\right))\end{array}\end{array}$$

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Mathematica [B]  time = 0.25, size = 112, normalized size = 4.48 $$\begin{array}{c}\frac{1}{2}x(-4{\log }^2(2)+{\log }^2(4)+\log \left(x^2\right)(2x^2+2{\log }^2(2)-x\log (16)+(-4x+\log (16)){\log }^2\left(x^3\right)+2{\log }^4\left(x^3\right))-8x^2\log (\log (3))-8{\log }^2(2)\log (\log (3))+x\log (65536)\log (\log (3))+(16x-\log (65536)){\log }^2\left(x^3\right)\log (\log (3))-8{\log }^4\left(x^3\right)\log (\log (3)))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[2*x^2 - 4*x*Log[2] + 2*Log[2]^2 + (3*x^2 - 4*x*Log[2] + Log[2]^2)*Log[x^2/Log[3]^4] + (-12*x + 12*Lo
g[2])*Log[x^3]*Log[x^2/Log[3]^4] + 12*Log[x^3]^3*Log[x^2/Log[3]^4] + Log[x^3]^4*(2 + Log[x^2/Log[3]^4]) + Log[
x^3]^2*(-4*x + 4*Log[2] + (-4*x + 2*Log[2])*Log[x^2/Log[3]^4]),x]

[Out]

(x*(-4*Log[2]^2 + Log[4]^2 + Log[x^2]*(2*x^2 + 2*Log[2]^2 - x*Log[16] + (-4*x + Log[16])*Log[x^3]^2 + 2*Log[x^
3]^4) - 8*x^2*Log[Log[3]] - 8*Log[2]^2*Log[Log[3]] + x*Log[65536]*Log[Log[3]] + (16*x - Log[65536])*Log[x^3]^2
*Log[Log[3]] - 8*Log[x^3]^4*Log[Log[3]]))/2

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fricas [B]  time = 0.60, size = 192, normalized size = 7.68 $$\begin{array}{c}\frac{1}{16}x\log {(\log (3{)}^{12})}^4\log \left(\frac{x^2}{\log (3{)}^4}\right)+\frac{3}{4}x\log {(\log (3{)}^{12})}^3\log {\left(\frac{x^2}{\log (3{)}^4}\right)}^2+\frac{81}{16}x\log {\left(\frac{x^2}{\log (3{)}^4}\right)}^5-\frac{9}{2}(x^2-x\log (2))\log {\left(\frac{x^2}{\log (3{)}^4}\right)}^3+\frac{1}{8}(27x\log {\left(\frac{x^2}{\log (3{)}^4}\right)}^3-4(x^2-x\log (2))\log \left(\frac{x^2}{\log (3{)}^4}\right))\log {(\log (3{)}^{12})}^2+\frac{3}{4}(9x\log {\left(\frac{x^2}{\log (3{)}^4}\right)}^4-4(x^2-x\log (2))\log {\left(\frac{x^2}{\log (3{)}^4}\right)}^2)\log (\log (3{)}^{12})+(x^3-2x^2\log (2)+x\log (2{)}^2)\log \left(\frac{x^2}{\log (3{)}^4}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2/log(3)^4)+2)*log(x^3)^4+12*log(x^2/log(3)^4)*log(x^3)^3+((2*log(2)-4*x)*log(x^2/log(3)^4)+4
*log(2)-4*x)*log(x^3)^2+(12*log(2)-12*x)*log(x^2/log(3)^4)*log(x^3)+(log(2)^2-4*x*log(2)+3*x^2)*log(x^2/log(3)
^4)+2*log(2)^2-4*x*log(2)+2*x^2,x, algorithm="fricas")

[Out]

1/16*x*log(log(3)^12)^4*log(x^2/log(3)^4) + 3/4*x*log(log(3)^12)^3*log(x^2/log(3)^4)^2 + 81/16*x*log(x^2/log(3
)^4)^5 - 9/2*(x^2 - x*log(2))*log(x^2/log(3)^4)^3 + 1/8*(27*x*log(x^2/log(3)^4)^3 - 4*(x^2 - x*log(2))*log(x^2
/log(3)^4))*log(log(3)^12)^2 + 3/4*(9*x*log(x^2/log(3)^4)^4 - 4*(x^2 - x*log(2))*log(x^2/log(3)^4)^2)*log(log(
3)^12) + (x^3 - 2*x^2*log(2) + x*log(2)^2)*log(x^2/log(3)^4)

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giac [B]  time = 0.25, size = 114, normalized size = 4.56 $$\begin{array}{c}-324x(\log (\log (3))+2)\log (x{)}^4+162x\log (x{)}^5+648x\log (x{)}^4-36(x^2-x\log (2))\log (x{)}^3+36(x^2(2\log (\log (3))+1)-2(\log (2)\log (\log (3))+\log (2))x)\log (x{)}^2-36(x^2-2x\log (2))\log (x{)}^2+(x^3-2x^2\log (2)+x\log (2{)}^2)\log \left(\frac{x^2}{\log (3{)}^4}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2/log(3)^4)+2)*log(x^3)^4+12*log(x^2/log(3)^4)*log(x^3)^3+((2*log(2)-4*x)*log(x^2/log(3)^4)+4
*log(2)-4*x)*log(x^3)^2+(12*log(2)-12*x)*log(x^2/log(3)^4)*log(x^3)+(log(2)^2-4*x*log(2)+3*x^2)*log(x^2/log(3)
^4)+2*log(2)^2-4*x*log(2)+2*x^2,x, algorithm="giac")

[Out]

-324*x*(log(log(3)) + 2)*log(x)^4 + 162*x*log(x)^5 + 648*x*log(x)^4 - 36*(x^2 - x*log(2))*log(x)^3 + 36*(x^2*(
2*log(log(3)) + 1) - 2*(log(2)*log(log(3)) + log(2))*x)*log(x)^2 - 36*(x^2 - 2*x*log(2))*log(x)^2 + (x^3 - 2*x
^2*log(2) + x*log(2)^2)*log(x^2/log(3)^4)

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maple [B]  time = 1.57, size = 141, normalized size = 5.64

method	result	size
default	$x^3\ln \left(x^2\right)-8x\ln (2)\ln (\ln (3))\ln {\left(x^3\right)}^2+2x\ln (2)\ln \left(x^2\right)\ln {\left(x^3\right)}^2-4x^3\ln (\ln (3))+\ln (2{)}^2\ln \left(x^2\right)x-4\ln (2{)}^2\ln (\ln (3))x-2\ln (2)x^2\ln \left(x^2\right)+8x^2\ln (2)\ln (\ln (3))+\ln {\left(x^3\right)}^4\ln \left(x^2\right)x-4\ln (\ln (3))x\ln {\left(x^3\right)}^4-2x^2\ln \left(x^2\right)\ln {\left(x^3\right)}^2+8x^2\ln (\ln (3))\ln {\left(x^3\right)}^2$	$141$
risch	$\text{Expression too large to display}$	$16204$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x^2/ln(3)^4)+2)*ln(x^3)^4+12*ln(x^2/ln(3)^4)*ln(x^3)^3+((2*ln(2)-4*x)*ln(x^2/ln(3)^4)+4*ln(2)-4*x)*ln(
x^3)^2+(12*ln(2)-12*x)*ln(x^2/ln(3)^4)*ln(x^3)+(ln(2)^2-4*x*ln(2)+3*x^2)*ln(x^2/ln(3)^4)+2*ln(2)^2-4*x*ln(2)+2
*x^2,x,method=_RETURNVERBOSE)

[Out]

x^3*ln(x^2)-8*x*ln(2)*ln(ln(3))*ln(x^3)^2+2*x*ln(2)*ln(x^2)*ln(x^3)^2-4*x^3*ln(ln(3))+ln(2)^2*ln(x^2)*x-4*ln(2
)^2*ln(ln(3))*x-2*ln(2)*x^2*ln(x^2)+8*x^2*ln(2)*ln(ln(3))+ln(x^3)^4*ln(x^2)*x-4*ln(ln(3))*x*ln(x^3)^4-2*x^2*ln
(x^2)*ln(x^3)^2+8*x^2*ln(ln(3))*ln(x^3)^2

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maxima [B]  time = 0.48, size = 71, normalized size = 2.84 $$\begin{array}{c}x\log {\left(x^3\right)}^4\log \left(\frac{x^2}{\log (3{)}^4}\right)-2(x^2-x\log (2))\log {\left(x^3\right)}^2\log \left(\frac{x^2}{\log (3{)}^4}\right)+(x^3-2x^2\log (2)+x\log (2{)}^2)\log \left(\frac{x^2}{\log (3{)}^4}\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2/log(3)^4)+2)*log(x^3)^4+12*log(x^2/log(3)^4)*log(x^3)^3+((2*log(2)-4*x)*log(x^2/log(3)^4)+4
*log(2)-4*x)*log(x^3)^2+(12*log(2)-12*x)*log(x^2/log(3)^4)*log(x^3)+(log(2)^2-4*x*log(2)+3*x^2)*log(x^2/log(3)
^4)+2*log(2)^2-4*x*log(2)+2*x^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 $$\begin{array}{c}\int 12{\ln \left(x^3\right)}^3\ln \left(\frac{x^2}{{\ln (3)}^4}\right)-4x\ln (2)+\ln \left(\frac{x^2}{{\ln (3)}^4}\right)(3x^2-4\ln (2)x+{\ln (2)}^2)+2{\ln (2)}^2+{\ln \left(x^3\right)}^4(\ln \left(\frac{x^2}{{\ln (3)}^4}\right)+2)-{\ln \left(x^3\right)}^2(4x-4\ln (2)+\ln \left(\frac{x^2}{{\ln (3)}^4}\right)(4x-2\ln (2)))+2x^2-\ln \left(x^3\right)\ln \left(\frac{x^2}{{\ln (3)}^4}\right)(12x-12\ln (2))dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(12*log(x^3)^3*log(x^2/log(3)^4) - 4*x*log(2) + log(x^2/log(3)^4)*(log(2)^2 - 4*x*log(2) + 3*x^2) + 2*log(2
)^2 + log(x^3)^4*(log(x^2/log(3)^4) + 2) - log(x^3)^2*(4*x - 4*log(2) + log(x^2/log(3)^4)*(4*x - 2*log(2))) +
2*x^2 - log(x^3)*log(x^2/log(3)^4)*(12*x - 12*log(2)),x)

[Out]

int(12*log(x^3)^3*log(x^2/log(3)^4) - 4*x*log(2) + log(x^2/log(3)^4)*(log(2)^2 - 4*x*log(2) + 3*x^2) + 2*log(2
)^2 + log(x^3)^4*(log(x^2/log(3)^4) + 2) - log(x^3)^2*(4*x - 4*log(2) + log(x^2/log(3)^4)*(4*x - 2*log(2))) +
2*x^2 - log(x^3)*log(x^2/log(3)^4)*(12*x - 12*log(2)), x)

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sympy [B]  time = 0.42, size = 146, normalized size = 5.84 $$\begin{array}{c}-4x^3\log (\log (3))+8x^2\log (2)\log (\log (3))+\frac{2x\log {\left(x^3\right)}^5}{3}-4x\log {\left(x^3\right)}^4\log (\log (3))-4x\log {(2)}^2\log (\log (3))+(-\frac{4x^2}{3}+\frac{4x\log (2)}{3})\log {\left(x^3\right)}^3+(8x^2\log (\log (3))-8x\log (2)\log (\log (3)))\log {\left(x^3\right)}^2+(\frac{2x^3}{3}-\frac{4x^2\log (2)}{3}+\frac{2x\log {(2)}^2}{3})\log \left(x^3\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x**2/ln(3)**4)+2)*ln(x**3)**4+12*ln(x**2/ln(3)**4)*ln(x**3)**3+((2*ln(2)-4*x)*ln(x**2/ln(3)**4)+
4*ln(2)-4*x)*ln(x**3)**2+(12*ln(2)-12*x)*ln(x**2/ln(3)**4)*ln(x**3)+(ln(2)**2-4*x*ln(2)+3*x**2)*ln(x**2/ln(3)*
*4)+2*ln(2)**2-4*x*ln(2)+2*x**2,x)

[Out]

-4*x**3*log(log(3)) + 8*x**2*log(2)*log(log(3)) + 2*x*log(x**3)**5/3 - 4*x*log(x**3)**4*log(log(3)) - 4*x*log(
2)**2*log(log(3)) + (-4*x**2/3 + 4*x*log(2)/3)*log(x**3)**3 + (8*x**2*log(log(3)) - 8*x*log(2)*log(log(3)))*lo
g(x**3)**2 + (2*x**3/3 - 4*x**2*log(2)/3 + 2*x*log(2)**2/3)*log(x**3)

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