3.20.18 \(\int \frac {-78732+769826 x+1168020 x^2+707476 x^3+231100 x^4+44308 x^5+5000 x^6+308 x^7+8 x^8+(8748-89100 x-93804 x^2-37092 x^3-7092 x^4-660 x^5-24 x^6) \log (\frac {x}{3})+(-324+3432 x+2088 x^2+396 x^3+24 x^4) \log ^2(\frac {x}{3})+(4-44 x-8 x^2) \log ^3(\frac {x}{3})+(-34992+356400 x+375216 x^2+148368 x^3+28368 x^4+2640 x^5+96 x^6+(2592-27456 x-16704 x^2-3168 x^3-192 x^4) \log (\frac {x}{3})+(-48+528 x+96 x^2) \log ^2(\frac {x}{3})) \log (\log (3))+(-5184+54912 x+33408 x^2+6336 x^3+384 x^4+(192-2112 x-384 x^2) \log (\frac {x}{3})) \log ^2(\log (3))+(-256+2816 x+512 x^2) \log ^3(\log (3))}{x} \, dx\)

Optimal. Leaf size=29 \[ 2 x+\left (-2-x-(5+x)^2+\log \left (\frac {x}{3}\right )-4 \log (\log (3))\right )^4 \]

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Rubi [B]  time = 0.92, antiderivative size = 599, normalized size of antiderivative = 20.66, number of steps used = 35, number of rules used = 9, integrand size = 268, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {14, 2357, 2296, 2295, 2302, 30, 2305, 2304, 2301} \begin {gather*} x^8+44 x^7+\frac {2 x^6}{3}-4 x^6 \log \left (\frac {x}{3}\right )+\frac {4}{3} x^6 (625+12 \log (\log (3)))+\frac {132 x^5}{5}-132 x^5 \log \left (\frac {x}{3}\right )+\frac {44}{5} x^5 (1007+60 \log (\log (3)))+\frac {3 x^4}{4}+6 x^4 \log ^2\left (\frac {x}{3}\right )+x^4 \left (57775+96 \log ^2(\log (3))+7092 \log (\log (3))\right )-3 x^4 \log \left (\frac {x}{3}\right )-3 x^4 (591+16 \log (\log (3))) \log \left (\frac {x}{3}\right )+\frac {3}{4} x^4 (591+16 \log (\log (3)))+\frac {88 x^3}{3}+132 x^3 \log ^2\left (\frac {x}{3}\right )+\frac {44}{3} x^3 \left (16079+144 \log ^2(\log (3))+3372 \log (\log (3))\right )-88 x^3 \log \left (\frac {x}{3}\right )-44 x^3 (281+24 \log (\log (3))) \log \left (\frac {x}{3}\right )+\frac {44}{3} x^3 (281+24 \log (\log (3)))+3 x^2-4 x^2 \log ^3\left (\frac {x}{3}\right )+6 x^2 \log ^2\left (\frac {x}{3}\right )-6 x^2 \left (7817+32 \log ^2(\log (3))+1392 \log (\log (3))\right ) \log \left (\frac {x}{3}\right )+3 x^2 \left (7817+32 \log ^2(\log (3))+1392 \log (\log (3))\right )+12 x^2 (87+4 \log (\log (3))) \log ^2\left (\frac {x}{3}\right )+2 x^2 \left (292005+128 \log ^3(\log (3))+8352 \log ^2(\log (3))+93804 \log (\log (3))\right )-6 x^2 \log \left (\frac {x}{3}\right )-12 x^2 (87+4 \log (\log (3))) \log \left (\frac {x}{3}\right )+6 x^2 (87+4 \log (\log (3)))+264 x+\log ^4\left (\frac {x}{3}\right )-44 x \log ^3\left (\frac {x}{3}\right )-4 (27+4 \log (\log (3))) \log ^3\left (\frac {x}{3}\right )+132 x \log ^2\left (\frac {x}{3}\right )-132 x \left (675+16 \log ^2(\log (3))+208 \log (\log (3))\right ) \log \left (\frac {x}{3}\right )+132 x \left (675+16 \log ^2(\log (3))+208 \log (\log (3))\right )+264 x (13+2 \log (\log (3))) \log ^2\left (\frac {x}{3}\right )+6 (27+4 \log (\log (3)))^2 \log ^2\left (\frac {x}{3}\right )+2 x \left (384913+1408 \log ^3(\log (3))+27456 \log ^2(\log (3))+178200 \log (\log (3))\right )-264 x \log \left (\frac {x}{3}\right )-528 x (13+2 \log (\log (3))) \log \left (\frac {x}{3}\right )+528 x (13+2 \log (\log (3)))-4 (27+4 \log (\log (3)))^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-78732 + 769826*x + 1168020*x^2 + 707476*x^3 + 231100*x^4 + 44308*x^5 + 5000*x^6 + 308*x^7 + 8*x^8 + (874
8 - 89100*x - 93804*x^2 - 37092*x^3 - 7092*x^4 - 660*x^5 - 24*x^6)*Log[x/3] + (-324 + 3432*x + 2088*x^2 + 396*
x^3 + 24*x^4)*Log[x/3]^2 + (4 - 44*x - 8*x^2)*Log[x/3]^3 + (-34992 + 356400*x + 375216*x^2 + 148368*x^3 + 2836
8*x^4 + 2640*x^5 + 96*x^6 + (2592 - 27456*x - 16704*x^2 - 3168*x^3 - 192*x^4)*Log[x/3] + (-48 + 528*x + 96*x^2
)*Log[x/3]^2)*Log[Log[3]] + (-5184 + 54912*x + 33408*x^2 + 6336*x^3 + 384*x^4 + (192 - 2112*x - 384*x^2)*Log[x
/3])*Log[Log[3]]^2 + (-256 + 2816*x + 512*x^2)*Log[Log[3]]^3)/x,x]

[Out]

264*x + 3*x^2 + (88*x^3)/3 + (3*x^4)/4 + (132*x^5)/5 + (2*x^6)/3 + 44*x^7 + x^8 - 264*x*Log[x/3] - 6*x^2*Log[x
/3] - 88*x^3*Log[x/3] - 3*x^4*Log[x/3] - 132*x^5*Log[x/3] - 4*x^6*Log[x/3] + 132*x*Log[x/3]^2 + 6*x^2*Log[x/3]
^2 + 132*x^3*Log[x/3]^2 + 6*x^4*Log[x/3]^2 - 44*x*Log[x/3]^3 - 4*x^2*Log[x/3]^3 + Log[x/3]^4 + 528*x*(13 + 2*L
og[Log[3]]) - 528*x*Log[x/3]*(13 + 2*Log[Log[3]]) + 264*x*Log[x/3]^2*(13 + 2*Log[Log[3]]) - 4*Log[x/3]^3*(27 +
4*Log[Log[3]]) + 6*Log[x/3]^2*(27 + 4*Log[Log[3]])^2 - 4*Log[x]*(27 + 4*Log[Log[3]])^3 + 6*x^2*(87 + 4*Log[Lo
g[3]]) - 12*x^2*Log[x/3]*(87 + 4*Log[Log[3]]) + 12*x^2*Log[x/3]^2*(87 + 4*Log[Log[3]]) + (4*x^6*(625 + 12*Log[
Log[3]]))/3 + (3*x^4*(591 + 16*Log[Log[3]]))/4 - 3*x^4*Log[x/3]*(591 + 16*Log[Log[3]]) + (44*x^3*(281 + 24*Log
[Log[3]]))/3 - 44*x^3*Log[x/3]*(281 + 24*Log[Log[3]]) + (44*x^5*(1007 + 60*Log[Log[3]]))/5 + 132*x*(675 + 208*
Log[Log[3]] + 16*Log[Log[3]]^2) - 132*x*Log[x/3]*(675 + 208*Log[Log[3]] + 16*Log[Log[3]]^2) + 3*x^2*(7817 + 13
92*Log[Log[3]] + 32*Log[Log[3]]^2) - 6*x^2*Log[x/3]*(7817 + 1392*Log[Log[3]] + 32*Log[Log[3]]^2) + x^4*(57775
+ 7092*Log[Log[3]] + 96*Log[Log[3]]^2) + (44*x^3*(16079 + 3372*Log[Log[3]] + 144*Log[Log[3]]^2))/3 + 2*x^2*(29
2005 + 93804*Log[Log[3]] + 8352*Log[Log[3]]^2 + 128*Log[Log[3]]^3) + 2*x*(384913 + 178200*Log[Log[3]] + 27456*
Log[Log[3]]^2 + 1408*Log[Log[3]]^3)

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 30

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

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 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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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 2357

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 \left (-1+11 x+2 x^2\right ) \log ^3\left (\frac {x}{3}\right )}{x}+\frac {12 \left (-1+11 x+2 x^2\right ) \log ^2\left (\frac {x}{3}\right ) \left (27+11 x+x^2+4 \log (\log (3))\right )}{x}-\frac {12 \left (-1+11 x+2 x^2\right ) \log \left (\frac {x}{3}\right ) \left (27+11 x+x^2+4 \log (\log (3))\right )^2}{x}+\frac {2 \left (154 x^7+4 x^8+2500 x^6 \left (1+\frac {12}{625} \log (\log (3))\right )+22154 x^5 \left (1+\frac {60 \log (\log (3))}{1007}\right )+115550 x^4 \left (1+\frac {12 \log (\log (3)) (591+8 \log (\log (3)))}{57775}\right )+353738 x^3 \left (1+\frac {12 \log (\log (3)) (281+12 \log (\log (3)))}{16079}\right )+384913 x \left (1+\frac {88 \log (\log (3)) \left (2025+312 \log (\log (3))+16 \log ^2(\log (3))\right )}{384913}\right )+584010 x^2 \left (1+\frac {4 \log (\log (3)) \left (23451+2088 \log (\log (3))+32 \log ^2(\log (3))\right )}{292005}\right )-39366 \left (1+\frac {4}{9} \log (\log (3)) \left (1+\frac {4 \log (\log (3)) (81+4 \log (\log (3)))}{2187}\right )\right )\right )}{x}\right ) \, dx\\ &=2 \int \frac {154 x^7+4 x^8+2500 x^6 \left (1+\frac {12}{625} \log (\log (3))\right )+22154 x^5 \left (1+\frac {60 \log (\log (3))}{1007}\right )+115550 x^4 \left (1+\frac {12 \log (\log (3)) (591+8 \log (\log (3)))}{57775}\right )+353738 x^3 \left (1+\frac {12 \log (\log (3)) (281+12 \log (\log (3)))}{16079}\right )+384913 x \left (1+\frac {88 \log (\log (3)) \left (2025+312 \log (\log (3))+16 \log ^2(\log (3))\right )}{384913}\right )+584010 x^2 \left (1+\frac {4 \log (\log (3)) \left (23451+2088 \log (\log (3))+32 \log ^2(\log (3))\right )}{292005}\right )-39366 \left (1+\frac {4}{9} \log (\log (3)) \left (1+\frac {4 \log (\log (3)) (81+4 \log (\log (3)))}{2187}\right )\right )}{x} \, dx-4 \int \frac {\left (-1+11 x+2 x^2\right ) \log ^3\left (\frac {x}{3}\right )}{x} \, dx+12 \int \frac {\left (-1+11 x+2 x^2\right ) \log ^2\left (\frac {x}{3}\right ) \left (27+11 x+x^2+4 \log (\log (3))\right )}{x} \, dx-12 \int \frac {\left (-1+11 x+2 x^2\right ) \log \left (\frac {x}{3}\right ) \left (27+11 x+x^2+4 \log (\log (3))\right )^2}{x} \, dx\\ &=2 \int \left (384913+154 x^6+4 x^7+178200 \log (\log (3))+27456 \log ^2(\log (3))+1408 \log ^3(\log (3))+4 x^5 (625+12 \log (\log (3)))+22 x^4 (1007+60 \log (\log (3)))+2 x^3 \left (57775+7092 \log (\log (3))+96 \log ^2(\log (3))\right )+22 x^2 \left (16079+3372 \log (\log (3))+144 \log ^2(\log (3))\right )-\frac {2 \left (19683+8748 \log (\log (3))+1296 \log ^2(\log (3))+64 \log ^3(\log (3))\right )}{x}+2 x \left (292005+93804 \log (\log (3))+8352 \log ^2(\log (3))+128 \log ^3(\log (3))\right )\right ) \, dx-4 \int \left (11 \log ^3\left (\frac {x}{3}\right )-\frac {\log ^3\left (\frac {x}{3}\right )}{x}+2 x \log ^3\left (\frac {x}{3}\right )\right ) \, dx+12 \int \left (33 x^2 \log ^2\left (\frac {x}{3}\right )+2 x^3 \log ^2\left (\frac {x}{3}\right )+\frac {\log ^2\left (\frac {x}{3}\right ) (-27-4 \log (\log (3)))}{x}+22 \log ^2\left (\frac {x}{3}\right ) (13+2 \log (\log (3)))+2 x \log ^2\left (\frac {x}{3}\right ) (87+4 \log (\log (3)))\right ) \, dx-12 \int \left (55 x^4 \log \left (\frac {x}{3}\right )+2 x^5 \log \left (\frac {x}{3}\right )+x^3 \log \left (\frac {x}{3}\right ) (591+16 \log (\log (3)))+11 x^2 \log \left (\frac {x}{3}\right ) (281+24 \log (\log (3)))+\frac {\log \left (\frac {x}{3}\right ) \left (-729-216 \log (\log (3))-16 \log ^2(\log (3))\right )}{x}+11 \log \left (\frac {x}{3}\right ) \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )+x \log \left (\frac {x}{3}\right ) \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )\right ) \, dx\\ &=44 x^7+x^8-4 \log (x) (27+4 \log (\log (3)))^3+\frac {4}{3} x^6 (625+12 \log (\log (3)))+\frac {44}{5} x^5 (1007+60 \log (\log (3)))+x^4 \left (57775+7092 \log (\log (3))+96 \log ^2(\log (3))\right )+\frac {44}{3} x^3 \left (16079+3372 \log (\log (3))+144 \log ^2(\log (3))\right )+2 x^2 \left (292005+93804 \log (\log (3))+8352 \log ^2(\log (3))+128 \log ^3(\log (3))\right )+2 x \left (384913+178200 \log (\log (3))+27456 \log ^2(\log (3))+1408 \log ^3(\log (3))\right )+4 \int \frac {\log ^3\left (\frac {x}{3}\right )}{x} \, dx-8 \int x \log ^3\left (\frac {x}{3}\right ) \, dx-24 \int x^5 \log \left (\frac {x}{3}\right ) \, dx+24 \int x^3 \log ^2\left (\frac {x}{3}\right ) \, dx-44 \int \log ^3\left (\frac {x}{3}\right ) \, dx+396 \int x^2 \log ^2\left (\frac {x}{3}\right ) \, dx-660 \int x^4 \log \left (\frac {x}{3}\right ) \, dx+(264 (13+2 \log (\log (3)))) \int \log ^2\left (\frac {x}{3}\right ) \, dx-(12 (27+4 \log (\log (3)))) \int \frac {\log ^2\left (\frac {x}{3}\right )}{x} \, dx+\left (12 (27+4 \log (\log (3)))^2\right ) \int \frac {\log \left (\frac {x}{3}\right )}{x} \, dx+(24 (87+4 \log (\log (3)))) \int x \log ^2\left (\frac {x}{3}\right ) \, dx-(12 (591+16 \log (\log (3)))) \int x^3 \log \left (\frac {x}{3}\right ) \, dx-(132 (281+24 \log (\log (3)))) \int x^2 \log \left (\frac {x}{3}\right ) \, dx-\left (132 \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )\right ) \int \log \left (\frac {x}{3}\right ) \, dx-\left (12 \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )\right ) \int x \log \left (\frac {x}{3}\right ) \, dx\\ &=\frac {132 x^5}{5}+\frac {2 x^6}{3}+44 x^7+x^8-132 x^5 \log \left (\frac {x}{3}\right )-4 x^6 \log \left (\frac {x}{3}\right )+132 x^3 \log ^2\left (\frac {x}{3}\right )+6 x^4 \log ^2\left (\frac {x}{3}\right )-44 x \log ^3\left (\frac {x}{3}\right )-4 x^2 \log ^3\left (\frac {x}{3}\right )+264 x \log ^2\left (\frac {x}{3}\right ) (13+2 \log (\log (3)))+6 \log ^2\left (\frac {x}{3}\right ) (27+4 \log (\log (3)))^2-4 \log (x) (27+4 \log (\log (3)))^3+12 x^2 \log ^2\left (\frac {x}{3}\right ) (87+4 \log (\log (3)))+\frac {4}{3} x^6 (625+12 \log (\log (3)))+\frac {3}{4} x^4 (591+16 \log (\log (3)))-3 x^4 \log \left (\frac {x}{3}\right ) (591+16 \log (\log (3)))+\frac {44}{3} x^3 (281+24 \log (\log (3)))-44 x^3 \log \left (\frac {x}{3}\right ) (281+24 \log (\log (3)))+\frac {44}{5} x^5 (1007+60 \log (\log (3)))+132 x \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )-132 x \log \left (\frac {x}{3}\right ) \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )+3 x^2 \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )-6 x^2 \log \left (\frac {x}{3}\right ) \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )+x^4 \left (57775+7092 \log (\log (3))+96 \log ^2(\log (3))\right )+\frac {44}{3} x^3 \left (16079+3372 \log (\log (3))+144 \log ^2(\log (3))\right )+2 x^2 \left (292005+93804 \log (\log (3))+8352 \log ^2(\log (3))+128 \log ^3(\log (3))\right )+2 x \left (384913+178200 \log (\log (3))+27456 \log ^2(\log (3))+1408 \log ^3(\log (3))\right )+4 \operatorname {Subst}\left (\int x^3 \, dx,x,\log \left (\frac {x}{3}\right )\right )-12 \int x^3 \log \left (\frac {x}{3}\right ) \, dx+12 \int x \log ^2\left (\frac {x}{3}\right ) \, dx+132 \int \log ^2\left (\frac {x}{3}\right ) \, dx-264 \int x^2 \log \left (\frac {x}{3}\right ) \, dx-(528 (13+2 \log (\log (3)))) \int \log \left (\frac {x}{3}\right ) \, dx-(12 (27+4 \log (\log (3)))) \operatorname {Subst}\left (\int x^2 \, dx,x,\log \left (\frac {x}{3}\right )\right )-(24 (87+4 \log (\log (3)))) \int x \log \left (\frac {x}{3}\right ) \, dx\\ &=\frac {88 x^3}{3}+\frac {3 x^4}{4}+\frac {132 x^5}{5}+\frac {2 x^6}{3}+44 x^7+x^8-88 x^3 \log \left (\frac {x}{3}\right )-3 x^4 \log \left (\frac {x}{3}\right )-132 x^5 \log \left (\frac {x}{3}\right )-4 x^6 \log \left (\frac {x}{3}\right )+132 x \log ^2\left (\frac {x}{3}\right )+6 x^2 \log ^2\left (\frac {x}{3}\right )+132 x^3 \log ^2\left (\frac {x}{3}\right )+6 x^4 \log ^2\left (\frac {x}{3}\right )-44 x \log ^3\left (\frac {x}{3}\right )-4 x^2 \log ^3\left (\frac {x}{3}\right )+\log ^4\left (\frac {x}{3}\right )+528 x (13+2 \log (\log (3)))-528 x \log \left (\frac {x}{3}\right ) (13+2 \log (\log (3)))+264 x \log ^2\left (\frac {x}{3}\right ) (13+2 \log (\log (3)))-4 \log ^3\left (\frac {x}{3}\right ) (27+4 \log (\log (3)))+6 \log ^2\left (\frac {x}{3}\right ) (27+4 \log (\log (3)))^2-4 \log (x) (27+4 \log (\log (3)))^3+6 x^2 (87+4 \log (\log (3)))-12 x^2 \log \left (\frac {x}{3}\right ) (87+4 \log (\log (3)))+12 x^2 \log ^2\left (\frac {x}{3}\right ) (87+4 \log (\log (3)))+\frac {4}{3} x^6 (625+12 \log (\log (3)))+\frac {3}{4} x^4 (591+16 \log (\log (3)))-3 x^4 \log \left (\frac {x}{3}\right ) (591+16 \log (\log (3)))+\frac {44}{3} x^3 (281+24 \log (\log (3)))-44 x^3 \log \left (\frac {x}{3}\right ) (281+24 \log (\log (3)))+\frac {44}{5} x^5 (1007+60 \log (\log (3)))+132 x \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )-132 x \log \left (\frac {x}{3}\right ) \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )+3 x^2 \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )-6 x^2 \log \left (\frac {x}{3}\right ) \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )+x^4 \left (57775+7092 \log (\log (3))+96 \log ^2(\log (3))\right )+\frac {44}{3} x^3 \left (16079+3372 \log (\log (3))+144 \log ^2(\log (3))\right )+2 x^2 \left (292005+93804 \log (\log (3))+8352 \log ^2(\log (3))+128 \log ^3(\log (3))\right )+2 x \left (384913+178200 \log (\log (3))+27456 \log ^2(\log (3))+1408 \log ^3(\log (3))\right )-12 \int x \log \left (\frac {x}{3}\right ) \, dx-264 \int \log \left (\frac {x}{3}\right ) \, dx\\ &=264 x+3 x^2+\frac {88 x^3}{3}+\frac {3 x^4}{4}+\frac {132 x^5}{5}+\frac {2 x^6}{3}+44 x^7+x^8-264 x \log \left (\frac {x}{3}\right )-6 x^2 \log \left (\frac {x}{3}\right )-88 x^3 \log \left (\frac {x}{3}\right )-3 x^4 \log \left (\frac {x}{3}\right )-132 x^5 \log \left (\frac {x}{3}\right )-4 x^6 \log \left (\frac {x}{3}\right )+132 x \log ^2\left (\frac {x}{3}\right )+6 x^2 \log ^2\left (\frac {x}{3}\right )+132 x^3 \log ^2\left (\frac {x}{3}\right )+6 x^4 \log ^2\left (\frac {x}{3}\right )-44 x \log ^3\left (\frac {x}{3}\right )-4 x^2 \log ^3\left (\frac {x}{3}\right )+\log ^4\left (\frac {x}{3}\right )+528 x (13+2 \log (\log (3)))-528 x \log \left (\frac {x}{3}\right ) (13+2 \log (\log (3)))+264 x \log ^2\left (\frac {x}{3}\right ) (13+2 \log (\log (3)))-4 \log ^3\left (\frac {x}{3}\right ) (27+4 \log (\log (3)))+6 \log ^2\left (\frac {x}{3}\right ) (27+4 \log (\log (3)))^2-4 \log (x) (27+4 \log (\log (3)))^3+6 x^2 (87+4 \log (\log (3)))-12 x^2 \log \left (\frac {x}{3}\right ) (87+4 \log (\log (3)))+12 x^2 \log ^2\left (\frac {x}{3}\right ) (87+4 \log (\log (3)))+\frac {4}{3} x^6 (625+12 \log (\log (3)))+\frac {3}{4} x^4 (591+16 \log (\log (3)))-3 x^4 \log \left (\frac {x}{3}\right ) (591+16 \log (\log (3)))+\frac {44}{3} x^3 (281+24 \log (\log (3)))-44 x^3 \log \left (\frac {x}{3}\right ) (281+24 \log (\log (3)))+\frac {44}{5} x^5 (1007+60 \log (\log (3)))+132 x \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )-132 x \log \left (\frac {x}{3}\right ) \left (675+208 \log (\log (3))+16 \log ^2(\log (3))\right )+3 x^2 \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )-6 x^2 \log \left (\frac {x}{3}\right ) \left (7817+1392 \log (\log (3))+32 \log ^2(\log (3))\right )+x^4 \left (57775+7092 \log (\log (3))+96 \log ^2(\log (3))\right )+\frac {44}{3} x^3 \left (16079+3372 \log (\log (3))+144 \log ^2(\log (3))\right )+2 x^2 \left (292005+93804 \log (\log (3))+8352 \log ^2(\log (3))+128 \log ^3(\log (3))\right )+2 x \left (384913+178200 \log (\log (3))+27456 \log ^2(\log (3))+1408 \log ^3(\log (3))\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.18, size = 235, normalized size = 8.10 \begin {gather*} \log ^4\left (\frac {x}{3}\right )-4 \log (x) (27+4 \log (\log (3)))^3-4 \log ^3\left (\frac {x}{3}\right ) \left (27+11 x+x^2+4 \log (\log (3))\right )+6 \log ^2\left (\frac {x}{3}\right ) \left (27+11 x+x^2+4 \log (\log (3))\right )^2-4 x (11+x) \log \left (\frac {x}{3}\right ) \left (22 x^3+x^4+33 x (27+4 \log (\log (3)))+3 (27+4 \log (\log (3)))^2+2 x^2 (101+6 \log (\log (3)))\right )+x \left (866054+44 x^6+x^7+384912 \log (\log (3))+57024 \log ^2(\log (3))+2816 \log ^3(\log (3))+88 x^4 (101+6 \log (\log (3)))+2 x^5 (417+8 \log (\log (3)))+2 x (27+4 \log (\log (3)))^2 (417+8 \log (\log (3)))+88 x^2 \left (2727+566 \log (\log (3))+24 \log ^2(\log (3))\right )+x^3 \left (58219+7104 \log (\log (3))+96 \log ^2(\log (3))\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-78732 + 769826*x + 1168020*x^2 + 707476*x^3 + 231100*x^4 + 44308*x^5 + 5000*x^6 + 308*x^7 + 8*x^8
+ (8748 - 89100*x - 93804*x^2 - 37092*x^3 - 7092*x^4 - 660*x^5 - 24*x^6)*Log[x/3] + (-324 + 3432*x + 2088*x^2
+ 396*x^3 + 24*x^4)*Log[x/3]^2 + (4 - 44*x - 8*x^2)*Log[x/3]^3 + (-34992 + 356400*x + 375216*x^2 + 148368*x^3
+ 28368*x^4 + 2640*x^5 + 96*x^6 + (2592 - 27456*x - 16704*x^2 - 3168*x^3 - 192*x^4)*Log[x/3] + (-48 + 528*x +
96*x^2)*Log[x/3]^2)*Log[Log[3]] + (-5184 + 54912*x + 33408*x^2 + 6336*x^3 + 384*x^4 + (192 - 2112*x - 384*x^2)
*Log[x/3])*Log[Log[3]]^2 + (-256 + 2816*x + 512*x^2)*Log[Log[3]]^3)/x,x]

[Out]

Log[x/3]^4 - 4*Log[x]*(27 + 4*Log[Log[3]])^3 - 4*Log[x/3]^3*(27 + 11*x + x^2 + 4*Log[Log[3]]) + 6*Log[x/3]^2*(
27 + 11*x + x^2 + 4*Log[Log[3]])^2 - 4*x*(11 + x)*Log[x/3]*(22*x^3 + x^4 + 33*x*(27 + 4*Log[Log[3]]) + 3*(27 +
4*Log[Log[3]])^2 + 2*x^2*(101 + 6*Log[Log[3]])) + x*(866054 + 44*x^6 + x^7 + 384912*Log[Log[3]] + 57024*Log[L
og[3]]^2 + 2816*Log[Log[3]]^3 + 88*x^4*(101 + 6*Log[Log[3]]) + 2*x^5*(417 + 8*Log[Log[3]]) + 2*x*(27 + 4*Log[L
og[3]])^2*(417 + 8*Log[Log[3]]) + 88*x^2*(2727 + 566*Log[Log[3]] + 24*Log[Log[3]]^2) + x^3*(58219 + 7104*Log[L
og[3]] + 96*Log[Log[3]]^2))

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fricas [B]  time = 0.75, size = 263, normalized size = 9.07 \begin {gather*} x^{8} + 44 \, x^{7} + 834 \, x^{6} + 8888 \, x^{5} + 58219 \, x^{4} - 4 \, {\left (x^{2} + 11 \, x + 27\right )} \log \left (\frac {1}{3} \, x\right )^{3} + \log \left (\frac {1}{3} \, x\right )^{4} + 256 \, {\left (x^{2} + 11 \, x - \log \left (\frac {1}{3} \, x\right )\right )} \log \left (\log \relax (3)\right )^{3} + 239976 \, x^{3} + 6 \, {\left (x^{4} + 22 \, x^{3} + 175 \, x^{2} + 594 \, x + 729\right )} \log \left (\frac {1}{3} \, x\right )^{2} + 96 \, {\left (x^{4} + 22 \, x^{3} + 175 \, x^{2} - 2 \, {\left (x^{2} + 11 \, x + 27\right )} \log \left (\frac {1}{3} \, x\right ) + \log \left (\frac {1}{3} \, x\right )^{2} + 594 \, x\right )} \log \left (\log \relax (3)\right )^{2} + 607986 \, x^{2} - 4 \, {\left (x^{6} + 33 \, x^{5} + 444 \, x^{4} + 3113 \, x^{3} + 11988 \, x^{2} + 24057 \, x + 19683\right )} \log \left (\frac {1}{3} \, x\right ) + 16 \, {\left (x^{6} + 33 \, x^{5} + 444 \, x^{4} + 3113 \, x^{3} + 3 \, {\left (x^{2} + 11 \, x + 27\right )} \log \left (\frac {1}{3} \, x\right )^{2} - \log \left (\frac {1}{3} \, x\right )^{3} + 11988 \, x^{2} - 3 \, {\left (x^{4} + 22 \, x^{3} + 175 \, x^{2} + 594 \, x + 729\right )} \log \left (\frac {1}{3} \, x\right ) + 24057 \, x\right )} \log \left (\log \relax (3)\right ) + 866054 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^2+2816*x-256)*log(log(3))^3+((-384*x^2-2112*x+192)*log(1/3*x)+384*x^4+6336*x^3+33408*x^2+549
12*x-5184)*log(log(3))^2+((96*x^2+528*x-48)*log(1/3*x)^2+(-192*x^4-3168*x^3-16704*x^2-27456*x+2592)*log(1/3*x)
+96*x^6+2640*x^5+28368*x^4+148368*x^3+375216*x^2+356400*x-34992)*log(log(3))+(-8*x^2-44*x+4)*log(1/3*x)^3+(24*
x^4+396*x^3+2088*x^2+3432*x-324)*log(1/3*x)^2+(-24*x^6-660*x^5-7092*x^4-37092*x^3-93804*x^2-89100*x+8748)*log(
1/3*x)+8*x^8+308*x^7+5000*x^6+44308*x^5+231100*x^4+707476*x^3+1168020*x^2+769826*x-78732)/x,x, algorithm="fric
as")

[Out]

x^8 + 44*x^7 + 834*x^6 + 8888*x^5 + 58219*x^4 - 4*(x^2 + 11*x + 27)*log(1/3*x)^3 + log(1/3*x)^4 + 256*(x^2 + 1
1*x - log(1/3*x))*log(log(3))^3 + 239976*x^3 + 6*(x^4 + 22*x^3 + 175*x^2 + 594*x + 729)*log(1/3*x)^2 + 96*(x^4
+ 22*x^3 + 175*x^2 - 2*(x^2 + 11*x + 27)*log(1/3*x) + log(1/3*x)^2 + 594*x)*log(log(3))^2 + 607986*x^2 - 4*(x
^6 + 33*x^5 + 444*x^4 + 3113*x^3 + 11988*x^2 + 24057*x + 19683)*log(1/3*x) + 16*(x^6 + 33*x^5 + 444*x^4 + 3113
*x^3 + 3*(x^2 + 11*x + 27)*log(1/3*x)^2 - log(1/3*x)^3 + 11988*x^2 - 3*(x^4 + 22*x^3 + 175*x^2 + 594*x + 729)*
log(1/3*x) + 24057*x)*log(log(3)) + 866054*x

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giac [B]  time = 1.81, size = 512, normalized size = 17.66 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^2+2816*x-256)*log(log(3))^3+((-384*x^2-2112*x+192)*log(1/3*x)+384*x^4+6336*x^3+33408*x^2+549
12*x-5184)*log(log(3))^2+((96*x^2+528*x-48)*log(1/3*x)^2+(-192*x^4-3168*x^3-16704*x^2-27456*x+2592)*log(1/3*x)
+96*x^6+2640*x^5+28368*x^4+148368*x^3+375216*x^2+356400*x-34992)*log(log(3))+(-8*x^2-44*x+4)*log(1/3*x)^3+(24*
x^4+396*x^3+2088*x^2+3432*x-324)*log(1/3*x)^2+(-24*x^6-660*x^5-7092*x^4-37092*x^3-93804*x^2-89100*x+8748)*log(
1/3*x)+8*x^8+308*x^7+5000*x^6+44308*x^5+231100*x^4+707476*x^3+1168020*x^2+769826*x-78732)/x,x, algorithm="giac
")

[Out]

x^8 + 44*x^7 + 2*x^6*(2*log(3) + 8*log(log(3)) + 417) + 44*x^5*(3*log(3) + 12*log(log(3)) + 202) + (6*log(3)^2
+ 48*log(3)*log(log(3)) + 96*log(log(3))^2 + 1776*log(3) + 7104*log(log(3)) + 58219)*x^4 + 44*(3*log(3)^2 + 2
4*log(3)*log(log(3)) + 48*log(log(3))^2 + 283*log(3) + 1132*log(log(3)) + 5454)*x^3 - 4*(x^2 + 11*x + log(3) +
4*log(log(3)) + 27)*log(x)^3 + log(x)^4 + 2*(2*log(3)^3 + 24*log(3)^2*log(log(3)) + 96*log(3)*log(log(3))^2 +
128*log(log(3))^3 + 525*log(3)^2 + 4200*log(3)*log(log(3)) + 8400*log(log(3))^2 + 23976*log(3) + 95904*log(lo
g(3)) + 303993)*x^2 + 6*(x^4 + 22*x^3 + x^2*(2*log(3) + 8*log(log(3)) + 175) + 22*x*(log(3) + 4*log(log(3)) +
27) + log(3)^2 + 8*log(3)*log(log(3)) + 16*log(log(3))^2 + 54*log(3) + 216*log(log(3)) + 729)*log(x)^2 + 2*(22
*log(3)^3 + 264*log(3)^2*log(log(3)) + 1056*log(3)*log(log(3))^2 + 1408*log(log(3))^3 + 1782*log(3)^2 + 14256*
log(3)*log(log(3)) + 28512*log(log(3))^2 + 48114*log(3) + 192456*log(log(3)) + 433027)*x - 4*(x^6 + 33*x^5 + 3
*x^4*(log(3) + 4*log(log(3)) + 148) + 11*x^3*(6*log(3) + 24*log(log(3)) + 283) + 3*(log(3)^2 + 8*log(3)*log(lo
g(3)) + 16*log(log(3))^2 + 175*log(3) + 700*log(log(3)) + 3996)*x^2 + 33*(log(3)^2 + 8*log(3)*log(log(3)) + 16
*log(log(3))^2 + 54*log(3) + 216*log(log(3)) + 729)*x)*log(x) - 4*(log(3)^3 + 12*log(3)^2*log(log(3)) + 48*log
(3)*log(log(3))^2 + 64*log(log(3))^3 + 81*log(3)^2 + 648*log(3)*log(log(3)) + 1296*log(log(3))^2 + 2187*log(3)
+ 8748*log(log(3)) + 19683)*log(x)

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maple [B]  time = 0.07, size = 332, normalized size = 11.45

method	result	size
risch	\(866054 x +44 x^{7}+x^{8}-78732 \ln \relax (x )+58219 x^{4}+239976 x^{3}+607986 x^{2}+834 x^{6}+8888 x^{5}+191808 x^{2} \ln \left (\ln \relax (3)\right )+49808 x^{3} \ln \left (\ln \relax (3)\right )+384912 \ln \left (\ln \relax (3)\right ) x -256 \ln \relax (x ) \ln \left (\ln \relax (3)\right )^{3}-5184 \ln \relax (x ) \ln \left (\ln \relax (3)\right )^{2}+\left (-4 x^{2}-16 \ln \left (\ln \relax (3)\right )-44 x -108\right ) \ln \left (\frac {x}{3}\right )^{3}+\left (6 x^{4}+48 x^{2} \ln \left (\ln \relax (3)\right )+132 x^{3}+96 \ln \left (\ln \relax (3)\right )^{2}+528 \ln \left (\ln \relax (3)\right ) x +1050 x^{2}+1296 \ln \left (\ln \relax (3)\right )+3564 x +4374\right ) \ln \left (\frac {x}{3}\right )^{2}+\left (-4 x^{6}-48 \ln \left (\ln \relax (3)\right ) x^{4}-132 x^{5}-192 \ln \left (\ln \relax (3)\right )^{2} x^{2}-1056 x^{3} \ln \left (\ln \relax (3)\right )-1776 x^{4}-2112 \ln \left (\ln \relax (3)\right )^{2} x -8400 x^{2} \ln \left (\ln \relax (3)\right )-12452 x^{3}-28512 \ln \left (\ln \relax (3)\right ) x -47952 x^{2}-96228 x \right ) \ln \left (\frac {x}{3}\right )+\ln \left (\frac {x}{3}\right )^{4}+7104 \ln \left (\ln \relax (3)\right ) x^{4}-34992 \ln \left (\ln \relax (3)\right ) \ln \relax (x )+16 \ln \left (\ln \relax (3)\right ) x^{6}+96 \ln \left (\ln \relax (3)\right )^{2} x^{4}+528 \ln \left (\ln \relax (3)\right ) x^{5}+256 \ln \left (\ln \relax (3)\right )^{3} x^{2}+2112 \ln \left (\ln \relax (3)\right )^{2} x^{3}+16800 \ln \left (\ln \relax (3)\right )^{2} x^{2}+57024 \ln \left (\ln \relax (3)\right )^{2} x +2816 \ln \left (\ln \relax (3)\right )^{3} x\)	\(332\)
derivativedivides	\(866054 x -47952 x^{2} \ln \left (\frac {x}{3}\right )-132 x^{5} \ln \left (\frac {x}{3}\right )-12452 x^{3} \ln \left (\frac {x}{3}\right )+3564 x \ln \left (\frac {x}{3}\right )^{2}-78732 \ln \left (\frac {x}{3}\right )+44 x^{7}+x^{8}+58219 x^{4}+239976 x^{3}+607986 x^{2}+834 x^{6}+8888 x^{5}+187608 x^{2} \ln \left (\ln \relax (3)\right )+1050 x^{2} \ln \left (\frac {x}{3}\right )^{2}+6 x^{4} \ln \left (\frac {x}{3}\right )^{2}-96228 x \ln \left (\frac {x}{3}\right )-1776 x^{4} \ln \left (\frac {x}{3}\right )+49456 x^{3} \ln \left (\ln \relax (3)\right )+356400 \ln \left (\ln \relax (3)\right ) x +132 x^{3} \ln \left (\frac {x}{3}\right )^{2}+4374 \ln \left (\frac {x}{3}\right )^{2}+\ln \left (\frac {x}{3}\right )^{4}+7092 \ln \left (\ln \relax (3)\right ) x^{4}-4 \ln \left (\frac {x}{3}\right ) x^{6}+16 \ln \left (\ln \relax (3)\right ) x^{6}+96 \ln \left (\ln \relax (3)\right )^{2} x^{4}+528 \ln \left (\ln \relax (3)\right ) x^{5}-4 \ln \left (\frac {x}{3}\right )^{3} x^{2}+256 \ln \left (\ln \relax (3)\right )^{3} x^{2}+2112 \ln \left (\ln \relax (3)\right )^{2} x^{3}+16704 \ln \left (\ln \relax (3)\right )^{2} x^{2}+1296 \ln \left (\frac {x}{3}\right )^{2} \ln \left (\ln \relax (3)\right )-5184 \ln \left (\frac {x}{3}\right ) \ln \left (\ln \relax (3)\right )^{2}+54912 \ln \left (\ln \relax (3)\right )^{2} x -34992 \ln \left (\frac {x}{3}\right ) \ln \left (\ln \relax (3)\right )-44 \ln \left (\frac {x}{3}\right )^{3} x +2816 \ln \left (\ln \relax (3)\right )^{3} x -16 \ln \left (\ln \relax (3)\right ) \ln \left (\frac {x}{3}\right )^{3}+96 \ln \left (\ln \relax (3)\right )^{2} \ln \left (\frac {x}{3}\right )^{2}-15552 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{4} \ln \left (\frac {x}{3}\right )}{324}-\frac {x^{4}}{1296}\right )+864 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{2} \ln \left (\frac {x}{3}\right )^{2}}{18}-\frac {x^{2} \ln \left (\frac {x}{3}\right )}{18}+\frac {x^{2}}{36}\right )-3456 \ln \left (\ln \relax (3)\right )^{2} \left (\frac {x^{2} \ln \left (\frac {x}{3}\right )}{18}-\frac {x^{2}}{36}\right )-85536 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{3} \ln \left (\frac {x}{3}\right )}{81}-\frac {x^{3}}{243}\right )-150336 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{2} \ln \left (\frac {x}{3}\right )}{18}-\frac {x^{2}}{36}\right )+1584 \ln \left (\ln \relax (3)\right ) \left (\frac {x \ln \left (\frac {x}{3}\right )^{2}}{3}-\frac {2 x \ln \left (\frac {x}{3}\right )}{3}+\frac {2 x}{3}\right )-6336 \ln \left (\ln \relax (3)\right )^{2} \left (\frac {x \ln \left (\frac {x}{3}\right )}{3}-\frac {x}{3}\right )-82368 \ln \left (\ln \relax (3)\right ) \left (\frac {x \ln \left (\frac {x}{3}\right )}{3}-\frac {x}{3}\right )-256 \ln \left (\ln \relax (3)\right )^{3} \ln \left (\frac {x}{3}\right )-108 \ln \left (\frac {x}{3}\right )^{3}\)	\(520\)
default	\(866054 x -47952 x^{2} \ln \left (\frac {x}{3}\right )-132 x^{5} \ln \left (\frac {x}{3}\right )-12452 x^{3} \ln \left (\frac {x}{3}\right )+3564 x \ln \left (\frac {x}{3}\right )^{2}-78732 \ln \left (\frac {x}{3}\right )+44 x^{7}+x^{8}+58219 x^{4}+239976 x^{3}+607986 x^{2}+834 x^{6}+8888 x^{5}+187608 x^{2} \ln \left (\ln \relax (3)\right )+1050 x^{2} \ln \left (\frac {x}{3}\right )^{2}+6 x^{4} \ln \left (\frac {x}{3}\right )^{2}-96228 x \ln \left (\frac {x}{3}\right )-1776 x^{4} \ln \left (\frac {x}{3}\right )+49456 x^{3} \ln \left (\ln \relax (3)\right )+356400 \ln \left (\ln \relax (3)\right ) x +132 x^{3} \ln \left (\frac {x}{3}\right )^{2}+4374 \ln \left (\frac {x}{3}\right )^{2}+\ln \left (\frac {x}{3}\right )^{4}+7092 \ln \left (\ln \relax (3)\right ) x^{4}-4 \ln \left (\frac {x}{3}\right ) x^{6}+16 \ln \left (\ln \relax (3)\right ) x^{6}+96 \ln \left (\ln \relax (3)\right )^{2} x^{4}+528 \ln \left (\ln \relax (3)\right ) x^{5}-4 \ln \left (\frac {x}{3}\right )^{3} x^{2}+256 \ln \left (\ln \relax (3)\right )^{3} x^{2}+2112 \ln \left (\ln \relax (3)\right )^{2} x^{3}+16704 \ln \left (\ln \relax (3)\right )^{2} x^{2}+1296 \ln \left (\frac {x}{3}\right )^{2} \ln \left (\ln \relax (3)\right )-5184 \ln \left (\frac {x}{3}\right ) \ln \left (\ln \relax (3)\right )^{2}+54912 \ln \left (\ln \relax (3)\right )^{2} x -34992 \ln \left (\frac {x}{3}\right ) \ln \left (\ln \relax (3)\right )-44 \ln \left (\frac {x}{3}\right )^{3} x +2816 \ln \left (\ln \relax (3)\right )^{3} x -16 \ln \left (\ln \relax (3)\right ) \ln \left (\frac {x}{3}\right )^{3}+96 \ln \left (\ln \relax (3)\right )^{2} \ln \left (\frac {x}{3}\right )^{2}-15552 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{4} \ln \left (\frac {x}{3}\right )}{324}-\frac {x^{4}}{1296}\right )+864 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{2} \ln \left (\frac {x}{3}\right )^{2}}{18}-\frac {x^{2} \ln \left (\frac {x}{3}\right )}{18}+\frac {x^{2}}{36}\right )-3456 \ln \left (\ln \relax (3)\right )^{2} \left (\frac {x^{2} \ln \left (\frac {x}{3}\right )}{18}-\frac {x^{2}}{36}\right )-85536 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{3} \ln \left (\frac {x}{3}\right )}{81}-\frac {x^{3}}{243}\right )-150336 \ln \left (\ln \relax (3)\right ) \left (\frac {x^{2} \ln \left (\frac {x}{3}\right )}{18}-\frac {x^{2}}{36}\right )+1584 \ln \left (\ln \relax (3)\right ) \left (\frac {x \ln \left (\frac {x}{3}\right )^{2}}{3}-\frac {2 x \ln \left (\frac {x}{3}\right )}{3}+\frac {2 x}{3}\right )-6336 \ln \left (\ln \relax (3)\right )^{2} \left (\frac {x \ln \left (\frac {x}{3}\right )}{3}-\frac {x}{3}\right )-82368 \ln \left (\ln \relax (3)\right ) \left (\frac {x \ln \left (\frac {x}{3}\right )}{3}-\frac {x}{3}\right )-256 \ln \left (\ln \relax (3)\right )^{3} \ln \left (\frac {x}{3}\right )-108 \ln \left (\frac {x}{3}\right )^{3}\)	\(520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((512*x^2+2816*x-256)*ln(ln(3))^3+((-384*x^2-2112*x+192)*ln(1/3*x)+384*x^4+6336*x^3+33408*x^2+54912*x-5184
)*ln(ln(3))^2+((96*x^2+528*x-48)*ln(1/3*x)^2+(-192*x^4-3168*x^3-16704*x^2-27456*x+2592)*ln(1/3*x)+96*x^6+2640*
x^5+28368*x^4+148368*x^3+375216*x^2+356400*x-34992)*ln(ln(3))+(-8*x^2-44*x+4)*ln(1/3*x)^3+(24*x^4+396*x^3+2088
*x^2+3432*x-324)*ln(1/3*x)^2+(-24*x^6-660*x^5-7092*x^4-37092*x^3-93804*x^2-89100*x+8748)*ln(1/3*x)+8*x^8+308*x
^7+5000*x^6+44308*x^5+231100*x^4+707476*x^3+1168020*x^2+769826*x-78732)/x,x,method=_RETURNVERBOSE)

[Out]

866054*x+44*x^7+x^8-78732*ln(x)+58219*x^4+239976*x^3+607986*x^2+834*x^6+8888*x^5+191808*x^2*ln(ln(3))+49808*x^
3*ln(ln(3))+384912*ln(ln(3))*x-256*ln(x)*ln(ln(3))^3-5184*ln(x)*ln(ln(3))^2+(-4*x^2-16*ln(ln(3))-44*x-108)*ln(
1/3*x)^3+(6*x^4+48*x^2*ln(ln(3))+132*x^3+96*ln(ln(3))^2+528*ln(ln(3))*x+1050*x^2+1296*ln(ln(3))+3564*x+4374)*l
n(1/3*x)^2+(-4*x^6-48*ln(ln(3))*x^4-132*x^5-192*ln(ln(3))^2*x^2-1056*x^3*ln(ln(3))-1776*x^4-2112*ln(ln(3))^2*x
-8400*x^2*ln(ln(3))-12452*x^3-28512*ln(ln(3))*x-47952*x^2-96228*x)*ln(1/3*x)+ln(1/3*x)^4+7104*ln(ln(3))*x^4-34
992*ln(ln(3))*ln(x)+16*ln(ln(3))*x^6+96*ln(ln(3))^2*x^4+528*ln(ln(3))*x^5+256*ln(ln(3))^3*x^2+2112*ln(ln(3))^2
*x^3+16800*ln(ln(3))^2*x^2+57024*ln(ln(3))^2*x+2816*ln(ln(3))^3*x

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maxima [B]  time = 0.62, size = 569, normalized size = 19.62 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x^2+2816*x-256)*log(log(3))^3+((-384*x^2-2112*x+192)*log(1/3*x)+384*x^4+6336*x^3+33408*x^2+549
12*x-5184)*log(log(3))^2+((96*x^2+528*x-48)*log(1/3*x)^2+(-192*x^4-3168*x^3-16704*x^2-27456*x+2592)*log(1/3*x)
+96*x^6+2640*x^5+28368*x^4+148368*x^3+375216*x^2+356400*x-34992)*log(log(3))+(-8*x^2-44*x+4)*log(1/3*x)^3+(24*
x^4+396*x^3+2088*x^2+3432*x-324)*log(1/3*x)^2+(-24*x^6-660*x^5-7092*x^4-37092*x^3-93804*x^2-89100*x+8748)*log(
1/3*x)+8*x^8+308*x^7+5000*x^6+44308*x^5+231100*x^4+707476*x^3+1168020*x^2+769826*x-78732)/x,x, algorithm="maxi
ma")

[Out]

x^8 + 44*x^7 - 4*x^6*log(1/3*x) + 16*x^6*log(log(3)) + 834*x^6 - 132*x^5*log(1/3*x) + 528*x^5*log(log(3)) + 96
*x^4*log(log(3))^2 + 3/4*(8*log(1/3*x)^2 - 4*log(1/3*x) + 1)*x^4 + 8888*x^5 - 1773*x^4*log(1/3*x) + 7092*x^4*l
og(log(3)) + 2112*x^3*log(log(3))^2 + 256*x^2*log(log(3))^3 + 44/3*(9*log(1/3*x)^2 - 6*log(1/3*x) + 2)*x^3 + 2
32873/4*x^4 - 12364*x^3*log(1/3*x) + log(1/3*x)^4 + 24*(2*log(1/3*x)^2 - 2*log(1/3*x) + 1)*x^2*log(log(3)) + 4
9456*x^3*log(log(3)) - 16*log(1/3*x)^3*log(log(3)) + 16704*x^2*log(log(3))^2 + 96*log(1/3*x)^2*log(log(3))^2 +
2816*x*log(log(3))^3 - 256*log(x)*log(log(3))^3 - (4*log(1/3*x)^3 - 6*log(1/3*x)^2 + 6*log(1/3*x) - 3)*x^2 +
522*(2*log(1/3*x)^2 - 2*log(1/3*x) + 1)*x^2 + 719840/3*x^3 - 46902*x^2*log(1/3*x) - 108*log(1/3*x)^3 + 528*(lo
g(1/3*x)^2 - 2*log(1/3*x) + 2)*x*log(log(3)) + 187608*x^2*log(log(3)) + 1296*log(1/3*x)^2*log(log(3)) - 96*(2*
x^2*log(1/3*x) - x^2)*log(log(3))^2 - 2112*(x*log(1/3*x) - x)*log(log(3))^2 + 54912*x*log(log(3))^2 - 5184*log
(x)*log(log(3))^2 - 44*(log(1/3*x)^3 - 3*log(1/3*x)^2 + 6*log(1/3*x) - 6)*x + 3432*(log(1/3*x)^2 - 2*log(1/3*x
) + 2)*x + 607461*x^2 - 89100*x*log(1/3*x) + 4374*log(1/3*x)^2 - 12*(4*x^4*log(1/3*x) - x^4)*log(log(3)) - 352
*(3*x^3*log(1/3*x) - x^3)*log(log(3)) - 4176*(2*x^2*log(1/3*x) - x^2)*log(log(3)) - 27456*(x*log(1/3*x) - x)*l
og(log(3)) + 356400*x*log(log(3)) - 34992*log(x)*log(log(3)) + 858926*x - 78732*log(x)

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mupad [B]  time = 2.02, size = 335, normalized size = 11.55 \begin {gather*} x^4\,\left (7104\,\ln \left (\ln \relax (3)\right )+96\,{\ln \left (\ln \relax (3)\right )}^2+58219\right )-\ln \relax (x)\,\left (34992\,\ln \left (\ln \relax (3)\right )+5184\,{\ln \left (\ln \relax (3)\right )}^2+256\,{\ln \left (\ln \relax (3)\right )}^3+78732\right )+x^3\,\left (49808\,\ln \left (\ln \relax (3)\right )+2112\,{\ln \left (\ln \relax (3)\right )}^2+239976\right )-44\,x\,{\ln \left (\frac {x}{3}\right )}^3-132\,x^5\,\ln \left (\frac {x}{3}\right )-4\,x^6\,\ln \left (\frac {x}{3}\right )-{\ln \left (\frac {x}{3}\right )}^3\,\left (16\,\ln \left (\ln \relax (3)\right )+108\right )+x\,\left (384912\,\ln \left (\ln \relax (3)\right )+57024\,{\ln \left (\ln \relax (3)\right )}^2+2816\,{\ln \left (\ln \relax (3)\right )}^3+866054\right )+{\ln \left (\frac {x}{3}\right )}^4+x^6\,\left (16\,\ln \left (\ln \relax (3)\right )+834\right )+x^5\,\left (528\,\ln \left (\ln \relax (3)\right )+8888\right )+44\,x^7+x^8-4\,x^2\,{\ln \left (\frac {x}{3}\right )}^3+132\,x^3\,{\ln \left (\frac {x}{3}\right )}^2+6\,x^4\,{\ln \left (\frac {x}{3}\right )}^2+6\,{\ln \left (\frac {x}{3}\right )}^2\,{\left (4\,\ln \left (\ln \relax (3)\right )+27\right )}^2-132\,x\,\ln \left (\frac {x}{3}\right )\,{\left (4\,\ln \left (\ln \relax (3)\right )+27\right )}^2-x^4\,\ln \left (\frac {x}{3}\right )\,\left (48\,\ln \left (\ln \relax (3)\right )+1776\right )+x\,{\ln \left (\frac {x}{3}\right )}^2\,\left (528\,\ln \left (\ln \relax (3)\right )+3564\right )-x^3\,\ln \left (\frac {x}{3}\right )\,\left (1056\,\ln \left (\ln \relax (3)\right )+12452\right )+x^2\,{\ln \left (\frac {x}{3}\right )}^2\,\left (48\,\ln \left (\ln \relax (3)\right )+1050\right )+2\,x^2\,{\left (4\,\ln \left (\ln \relax (3)\right )+27\right )}^2\,\left (8\,\ln \left (\ln \relax (3)\right )+417\right )-48\,x^2\,\ln \left (\frac {x}{3}\right )\,\left (\ln \left (\ln \relax (3)\right )+37\right )\,\left (4\,\ln \left (\ln \relax (3)\right )+27\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((769826*x + log(log(3))^3*(2816*x + 512*x^2 - 256) - log(x/3)^3*(44*x + 8*x^2 - 4) + log(log(3))*(356400*x
+ log(x/3)^2*(528*x + 96*x^2 - 48) - log(x/3)*(27456*x + 16704*x^2 + 3168*x^3 + 192*x^4 - 2592) + 375216*x^2
+ 148368*x^3 + 28368*x^4 + 2640*x^5 + 96*x^6 - 34992) + log(log(3))^2*(54912*x - log(x/3)*(2112*x + 384*x^2 -
192) + 33408*x^2 + 6336*x^3 + 384*x^4 - 5184) + 1168020*x^2 + 707476*x^3 + 231100*x^4 + 44308*x^5 + 5000*x^6 +
308*x^7 + 8*x^8 + log(x/3)^2*(3432*x + 2088*x^2 + 396*x^3 + 24*x^4 - 324) - log(x/3)*(89100*x + 93804*x^2 + 3
7092*x^3 + 7092*x^4 + 660*x^5 + 24*x^6 - 8748) - 78732)/x,x)

[Out]

x^4*(7104*log(log(3)) + 96*log(log(3))^2 + 58219) - log(x)*(34992*log(log(3)) + 5184*log(log(3))^2 + 256*log(l
og(3))^3 + 78732) + x^3*(49808*log(log(3)) + 2112*log(log(3))^2 + 239976) - 44*x*log(x/3)^3 - 132*x^5*log(x/3)
- 4*x^6*log(x/3) - log(x/3)^3*(16*log(log(3)) + 108) + x*(384912*log(log(3)) + 57024*log(log(3))^2 + 2816*log
(log(3))^3 + 866054) + log(x/3)^4 + x^6*(16*log(log(3)) + 834) + x^5*(528*log(log(3)) + 8888) + 44*x^7 + x^8 -
4*x^2*log(x/3)^3 + 132*x^3*log(x/3)^2 + 6*x^4*log(x/3)^2 + 6*log(x/3)^2*(4*log(log(3)) + 27)^2 - 132*x*log(x/
3)*(4*log(log(3)) + 27)^2 - x^4*log(x/3)*(48*log(log(3)) + 1776) + x*log(x/3)^2*(528*log(log(3)) + 3564) - x^3
*log(x/3)*(1056*log(log(3)) + 12452) + x^2*log(x/3)^2*(48*log(log(3)) + 1050) + 2*x^2*(4*log(log(3)) + 27)^2*(
8*log(log(3)) + 417) - 48*x^2*log(x/3)*(log(log(3)) + 37)*(4*log(log(3)) + 27)

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sympy [B]  time = 0.96, size = 333, normalized size = 11.48 \begin {gather*} x^{8} + 44 x^{7} + x^{6} \left (16 \log {\left (\log {\relax (3 )} \right )} + 834\right ) + x^{5} \left (528 \log {\left (\log {\relax (3 )} \right )} + 8888\right ) + x^{4} \left (96 \log {\left (\log {\relax (3 )} \right )}^{2} + 7104 \log {\left (\log {\relax (3 )} \right )} + 58219\right ) + x^{3} \left (2112 \log {\left (\log {\relax (3 )} \right )}^{2} + 49808 \log {\left (\log {\relax (3 )} \right )} + 239976\right ) + x^{2} \left (256 \log {\left (\log {\relax (3 )} \right )}^{3} + 16800 \log {\left (\log {\relax (3 )} \right )}^{2} + 191808 \log {\left (\log {\relax (3 )} \right )} + 607986\right ) + x \left (2816 \log {\left (\log {\relax (3 )} \right )}^{3} + 57024 \log {\left (\log {\relax (3 )} \right )}^{2} + 384912 \log {\left (\log {\relax (3 )} \right )} + 866054\right ) + \left (- 4 x^{2} - 44 x - 108 - 16 \log {\left (\log {\relax (3 )} \right )}\right ) \log {\left (\frac {x}{3} \right )}^{3} + \left (6 x^{4} + 132 x^{3} + 48 x^{2} \log {\left (\log {\relax (3 )} \right )} + 1050 x^{2} + 528 x \log {\left (\log {\relax (3 )} \right )} + 3564 x + 96 \log {\left (\log {\relax (3 )} \right )}^{2} + 1296 \log {\left (\log {\relax (3 )} \right )} + 4374\right ) \log {\left (\frac {x}{3} \right )}^{2} + \left (- 4 x^{6} - 132 x^{5} - 1776 x^{4} - 48 x^{4} \log {\left (\log {\relax (3 )} \right )} - 12452 x^{3} - 1056 x^{3} \log {\left (\log {\relax (3 )} \right )} - 47952 x^{2} - 8400 x^{2} \log {\left (\log {\relax (3 )} \right )} - 192 x^{2} \log {\left (\log {\relax (3 )} \right )}^{2} - 96228 x - 28512 x \log {\left (\log {\relax (3 )} \right )} - 2112 x \log {\left (\log {\relax (3 )} \right )}^{2}\right ) \log {\left (\frac {x}{3} \right )} + \log {\left (\frac {x}{3} \right )}^{4} - 4 \left (4 \log {\left (\log {\relax (3 )} \right )} + 27\right )^{3} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((512*x**2+2816*x-256)*ln(ln(3))**3+((-384*x**2-2112*x+192)*ln(1/3*x)+384*x**4+6336*x**3+33408*x**2+
54912*x-5184)*ln(ln(3))**2+((96*x**2+528*x-48)*ln(1/3*x)**2+(-192*x**4-3168*x**3-16704*x**2-27456*x+2592)*ln(1
/3*x)+96*x**6+2640*x**5+28368*x**4+148368*x**3+375216*x**2+356400*x-34992)*ln(ln(3))+(-8*x**2-44*x+4)*ln(1/3*x
)**3+(24*x**4+396*x**3+2088*x**2+3432*x-324)*ln(1/3*x)**2+(-24*x**6-660*x**5-7092*x**4-37092*x**3-93804*x**2-8
9100*x+8748)*ln(1/3*x)+8*x**8+308*x**7+5000*x**6+44308*x**5+231100*x**4+707476*x**3+1168020*x**2+769826*x-7873
2)/x,x)

[Out]

x**8 + 44*x**7 + x**6*(16*log(log(3)) + 834) + x**5*(528*log(log(3)) + 8888) + x**4*(96*log(log(3))**2 + 7104*
log(log(3)) + 58219) + x**3*(2112*log(log(3))**2 + 49808*log(log(3)) + 239976) + x**2*(256*log(log(3))**3 + 16
800*log(log(3))**2 + 191808*log(log(3)) + 607986) + x*(2816*log(log(3))**3 + 57024*log(log(3))**2 + 384912*log
(log(3)) + 866054) + (-4*x**2 - 44*x - 108 - 16*log(log(3)))*log(x/3)**3 + (6*x**4 + 132*x**3 + 48*x**2*log(lo
g(3)) + 1050*x**2 + 528*x*log(log(3)) + 3564*x + 96*log(log(3))**2 + 1296*log(log(3)) + 4374)*log(x/3)**2 + (-
4*x**6 - 132*x**5 - 1776*x**4 - 48*x**4*log(log(3)) - 12452*x**3 - 1056*x**3*log(log(3)) - 47952*x**2 - 8400*x
**2*log(log(3)) - 192*x**2*log(log(3))**2 - 96228*x - 28512*x*log(log(3)) - 2112*x*log(log(3))**2)*log(x/3) +
log(x/3)**4 - 4*(4*log(log(3)) + 27)**3*log(x)

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