3.19.80 \(\int \frac {(18+14 x^4+2 \log (3)) \log (\frac {x}{81-18 x^4+x^8+(18-2 x^4) \log (3)+\log ^2(3)})}{9 x-x^5+x \log (3)} \, dx\)

Optimal. Leaf size=18 \[ 4+\log ^2\left (\frac {x}{\left (-9+x^4-\log (3)\right )^2}\right ) \]

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Rubi [C]  time = 3.86, antiderivative size = 1299, normalized size of antiderivative = 72.17, number of steps used = 104, number of rules used = 16, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6, 1593, 2528, 2524, 2357, 2301, 2337, 2391, 2418, 2392, 260, 2416, 2390, 2394, 2393, 2315}

result too large to display

Antiderivative was successfully verified.

[In]

Int[((18 + 14*x^4 + 2*Log[3])*Log[x/(81 - 18*x^4 + x^8 + (18 - 2*x^4)*Log[3] + Log[3]^2)])/(9*x - x^5 + x*Log[
3]),x]

[Out]

-Log[x]^2 + 2*Log[x]*Log[9 + Log[3]] + 2*Log[x]*Log[x/(81 - 18*x^4 + x^8 + 2*(9 - x^4)*Log[3] + Log[3]^2)] + 4
*Log[x]*Log[1 - x^4/(9 + Log[3])] + 4*Log[((-I)*x)/(9 + Log[3])^(1/4)]*Log[-x + I*(9 + Log[3])^(1/4)] - 4*Log[
x/(81 - 18*x^4 + x^8 + 2*(9 - x^4)*Log[3] + Log[3]^2)]*Log[-x + I*(9 + Log[3])^(1/4)] - 8*Log[((-1/2 - I/2)*(x
- (9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[-x + I*(9 + Log[3])^(1/4)] - 4*Log[-x + I*(9 + Log[3])^(1/4)]^
2 + 4*Log[(I*x)/(9 + Log[3])^(1/4)]*Log[x + I*(9 + Log[3])^(1/4)] - 4*Log[x/(81 - 18*x^4 + x^8 + 2*(9 - x^4)*L
og[3] + Log[3]^2)]*Log[x + I*(9 + Log[3])^(1/4)] - 8*Log[((-1/2 + I/2)*(x - (9 + Log[3])^(1/4)))/(9 + Log[3])^
(1/4)]*Log[x + I*(9 + Log[3])^(1/4)] - 8*Log[((I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[x + I*
(9 + Log[3])^(1/4)] - 4*Log[x + I*(9 + Log[3])^(1/4)]^2 - 8*Log[-x + I*(9 + Log[3])^(1/4)]*Log[((-1/2*I)*(x +
I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] - 4*Log[x/(81 - 18*x^4 + x^8 + 2*(9 - x^4)*Log[3] + Log[3]^2)]*Log[
-x + (9 + Log[3])^(1/4)] - 8*Log[((1/2 + I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[-x + (9 + Lo
g[3])^(1/4)] - 8*Log[((1/2 - I/2)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[-x + (9 + Log[3])^(1/4)]
- 4*Log[-x + (9 + Log[3])^(1/4)]^2 - 4*Log[x/(81 - 18*x^4 + x^8 + 2*(9 - x^4)*Log[3] + Log[3]^2)]*Log[x + (9
+ Log[3])^(1/4)] - 8*Log[-1/2*(x - (9 + Log[3])^(1/4))/(9 + Log[3])^(1/4)]*Log[x + (9 + Log[3])^(1/4)] - 8*Log
[((-1/2 + I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[x + (9 + Log[3])^(1/4)] - 8*Log[((-1/2 - I/
2)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[x + (9 + Log[3])^(1/4)] - 4*Log[x + (9 + Log[3])^(1/4)]
^2 - 8*Log[-x + (9 + Log[3])^(1/4)]*Log[(x + (9 + Log[3])^(1/4))/(2*(9 + Log[3])^(1/4))] - 8*Log[-x + I*(9 + L
og[3])^(1/4)]*Log[((1/2 - I/2)*(x + (9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] - 8*Log[x + I*(9 + Log[3])^(1/4)]
*Log[((1/2 + I/2)*(x + (9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + PolyLog[2, x^4/(9 + Log[3])] - 4*PolyLog[2,
-(x/(9 + Log[3])^(1/4))] - 4*PolyLog[2, x/(9 + Log[3])^(1/4)] - 8*PolyLog[2, (-1/2 - I/2)*(I - x/(9 + Log[3])^
(1/4))] - 8*PolyLog[2, (1 - x/(9 + Log[3])^(1/4))/2] - 8*PolyLog[2, (1/2 - I/2)*(1 - x/(9 + Log[3])^(1/4))] -
8*PolyLog[2, (1/2 + I/2)*(1 - x/(9 + Log[3])^(1/4))] - 8*PolyLog[2, (1 - (I*x)/(9 + Log[3])^(1/4))/2] - 8*Poly
Log[2, (1/2 + I/2)*(1 - (I*x)/(9 + Log[3])^(1/4))] + 4*PolyLog[2, 1 - (I*x)/(9 + Log[3])^(1/4)] - 8*PolyLog[2,
(1 + (I*x)/(9 + Log[3])^(1/4))/2] - 8*PolyLog[2, (1/2 + I/2)*(1 + (I*x)/(9 + Log[3])^(1/4))] + 4*PolyLog[2, 1
+ (I*x)/(9 + Log[3])^(1/4)] - 8*PolyLog[2, (-1/2 - I/2)*(I + x/(9 + Log[3])^(1/4))] - 8*PolyLog[2, (1 + x/(9
+ Log[3])^(1/4))/2] - 8*PolyLog[2, (1/2 + I/2)*(1 + x/(9 + Log[3])^(1/4))] - 8*PolyLog[2, ((1/2 - I/2)*(x + (9
+ Log[3])^(1/4)))/(9 + Log[3])^(1/4)]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 260

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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 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 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

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]

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 2391

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

Rule 2392

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-x^5+x (9+\log (3))} \, dx\\ &=\int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x \left (9-x^4+\log (3)\right )} \, dx\\ &=\int \left (\frac {2 \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x}-\frac {16 x^3 \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-9+x^4-\log (3)}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x} \, dx-16 \int \frac {x^3 \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-9+x^4-\log (3)} \, dx\\ &=2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )-2 \int \frac {\left (81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)\right ) \left (-\frac {x \left (-72 x^3+8 x^7-8 x^3 \log (3)\right )}{\left (81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)\right )^2}+\frac {1}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right ) \log (x)}{x} \, dx-16 \int \left (\frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x^2-\sqrt {9+\log (3)}\right )}+\frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x^2+\sqrt {9+\log (3)}\right )}\right ) \, dx\\ &=2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )-2 \int \left (\frac {\log (x)}{x}-\frac {8 x^3 \log (x)}{-9+x^4-\log (3)}\right ) \, dx-8 \int \frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x^2-\sqrt {9+\log (3)}} \, dx-8 \int \frac {x \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x^2+\sqrt {9+\log (3)}} \, dx\\ &=2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )-2 \int \frac {\log (x)}{x} \, dx-8 \int \left (-\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (-x+i \sqrt [4]{9+\log (3)}\right )}+\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x+i \sqrt [4]{9+\log (3)}\right )}\right ) \, dx-8 \int \left (-\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (-x+\sqrt [4]{9+\log (3)}\right )}+\frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{2 \left (x+\sqrt [4]{9+\log (3)}\right )}\right ) \, dx+16 \int \frac {x^3 \log (x)}{-9+x^4-\log (3)} \, dx\\ &=-\log ^2(x)+2 \log (x) \log \left (\frac {x}{81-18 x^4+x^8+2 \left (9-x^4\right ) \log (3)+\log ^2(3)}\right )+4 \log (x) \log \left (1-\frac {x^4}{9+\log (3)}\right )-4 \int \frac {\log \left (1+\frac {x^4}{-9-\log (3)}\right )}{x} \, dx+4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-x+i \sqrt [4]{9+\log (3)}} \, dx-4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x+i \sqrt [4]{9+\log (3)}} \, dx+4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{-x+\sqrt [4]{9+\log (3)}} \, dx-4 \int \frac {\log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{x+\sqrt [4]{9+\log (3)}} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.52, size = 1485, normalized size = 82.50

result too large to display

Antiderivative was successfully verified.

[In]

Integrate[((18 + 14*x^4 + 2*Log[3])*Log[x/(81 - 18*x^4 + x^8 + (18 - 2*x^4)*Log[3] + Log[3]^2)])/(9*x - x^5 +
x*Log[3]),x]

[Out]

-2*(Log[x]^2/2 - (Log[x]*Log[9 + Log[3]])/2 - Log[x]*Log[x/(9 - x^4 + Log[3])^2] - 2*Log[-(x/(9 + Log[3])^(1/4
))]*Log[-x - (9 + Log[3])^(1/4)] + 4*Log[2*(9 + Log[3])^(1/4)]*Log[-x - (9 + Log[3])^(1/4)] + 2*Log[x/(9 - x^4
+ Log[3])^2]*Log[-x - (9 + Log[3])^(1/4)] + 2*Log[-x - (9 + Log[3])^(1/4)]^2 + 4*Log[-x - (9 + Log[3])^(1/4)]
*Log[-1/2*(x - (9 + Log[3])^(1/4))/(9 + Log[3])^(1/4)] - 2*Log[(I*x)/(9 + Log[3])^(1/4)]*Log[-x - I*(9 + Log[3
])^(1/4)] + 2*Log[x/(9 - x^4 + Log[3])^2]*Log[-x - I*(9 + Log[3])^(1/4)] + 4*Log[((-1/2 + I/2)*(x - (9 + Log[3
])^(1/4)))/(9 + Log[3])^(1/4)]*Log[-x - I*(9 + Log[3])^(1/4)] + 2*Log[-x - I*(9 + Log[3])^(1/4)]^2 + 4*Log[-x
- (9 + Log[3])^(1/4)]*Log[((-1/2 + I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*Log[-x - I*(9 + Lo
g[3])^(1/4)]*Log[((I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] - 2*Log[x]*Log[(I*(x - I*(9 + Log[3])^
(1/4)))/(9 + Log[3])^(1/4)] - 2*Log[((-I)*x)/(9 + Log[3])^(1/4)]*Log[-x + I*(9 + Log[3])^(1/4)] + 2*Log[x/(9 -
x^4 + Log[3])^2]*Log[-x + I*(9 + Log[3])^(1/4)] + 4*Log[((-1/2 - I/2)*(x - (9 + Log[3])^(1/4)))/(9 + Log[3])^
(1/4)]*Log[-x + I*(9 + Log[3])^(1/4)] + 2*Log[-x + I*(9 + Log[3])^(1/4)]^2 + 4*Log[-x - (9 + Log[3])^(1/4)]*Lo
g[((-1/2 - I/2)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*Log[-x + I*(9 + Log[3])^(1/4)]*Log[((-1/2*
I)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] - 2*Log[x]*Log[((-I)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3
])^(1/4)] - (Log[9 + Log[3]]*Log[-x + (9 + Log[3])^(1/4)])/2 + 2*Log[x/(9 - x^4 + Log[3])^2]*Log[-x + (9 + Log
[3])^(1/4)] + 4*Log[((1/2 + I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[-x + (9 + Log[3])^(1/4)]
+ 4*Log[((1/2 - I/2)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Log[-x + (9 + Log[3])^(1/4)] + 2*Log[-x +
(9 + Log[3])^(1/4)]^2 + 4*Log[-x + I*(9 + Log[3])^(1/4)]*Log[((1/2 - I/2)*(x + (9 + Log[3])^(1/4)))/(9 + Log[
3])^(1/4)] + 4*Log[-x - I*(9 + Log[3])^(1/4)]*Log[((1/2 + I/2)*(x + (9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] -
2*Log[x]*Log[(x + (9 + Log[3])^(1/4))/(9 + Log[3])^(1/4)] - 2*PolyLog[2, -(x/(9 + Log[3])^(1/4))] - 2*PolyLog
[2, ((-I)*x)/(9 + Log[3])^(1/4)] - 2*PolyLog[2, (I*x)/(9 + Log[3])^(1/4)] + 2*PolyLog[2, x/(9 + Log[3])^(1/4)]
+ 2*PolyLog[2, -((x - (9 + Log[3])^(1/4))/(9 + Log[3])^(1/4))] + 4*PolyLog[2, ((-1/2 - I/2)*(x - (9 + Log[3])
^(1/4)))/(9 + Log[3])^(1/4)] + 4*PolyLog[2, ((-1/2 + I/2)*(x - (9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*Po
lyLog[2, ((-1/2 + I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*PolyLog[2, ((I/2)*(x - I*(9 + Log[3
])^(1/4)))/(9 + Log[3])^(1/4)] - 2*PolyLog[2, (I*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*PolyLog[2
, ((1/2 + I/2)*(x - I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*PolyLog[2, ((-1/2 - I/2)*(x + I*(9 + Log[3]
)^(1/4)))/(9 + Log[3])^(1/4)] + 4*PolyLog[2, ((-1/2*I)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] - 2*Pol
yLog[2, ((-I)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*PolyLog[2, ((1/2 - I/2)*(x + I*(9 + Log[3])^
(1/4)))/(9 + Log[3])^(1/4)] + 4*PolyLog[2, ((1/2 - I/2)*(x + (9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] + 4*Poly
Log[2, ((1/2 + I/2)*(x + (9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)] - 2*PolyLog[2, (x + (9 + Log[3])^(1/4))/(9 +
Log[3])^(1/4)])

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fricas [A]  time = 0.93, size = 30, normalized size = 1.67 \begin {gather*} \log \left (\frac {x}{x^{8} - 18 \, x^{4} - 2 \, {\left (x^{4} - 9\right )} \log \relax (3) + \log \relax (3)^{2} + 81}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(3)+14*x^4+18)*log(x/(log(3)^2+(-2*x^4+18)*log(3)+x^8-18*x^4+81))/(x*log(3)-x^5+9*x),x, algori
thm="fricas")

[Out]

log(x/(x^8 - 18*x^4 - 2*(x^4 - 9)*log(3) + log(3)^2 + 81))^2

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giac [B]  time = 0.31, size = 64, normalized size = 3.56 \begin {gather*} 2 \, {\left (2 \, \log \left (x^{4} - \log \relax (3) - 9\right ) - \log \relax (x)\right )} \log \left (x^{8} - 2 \, x^{4} \log \relax (3) - 18 \, x^{4} + \log \relax (3)^{2} + 18 \, \log \relax (3) + 81\right ) - 4 \, \log \left (x^{4} - \log \relax (3) - 9\right )^{2} + \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(3)+14*x^4+18)*log(x/(log(3)^2+(-2*x^4+18)*log(3)+x^8-18*x^4+81))/(x*log(3)-x^5+9*x),x, algori
thm="giac")

[Out]

2*(2*log(x^4 - log(3) - 9) - log(x))*log(x^8 - 2*x^4*log(3) - 18*x^4 + log(3)^2 + 18*log(3) + 81) - 4*log(x^4
- log(3) - 9)^2 + log(x)^2

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maple [A]  time = 0.40, size = 32, normalized size = 1.78

method	result	size
norman	\(\ln \left (\frac {x}{\ln \relax (3)^{2}+\left (-2 x^{4}+18\right ) \ln \relax (3)+x^{8}-18 x^{4}+81}\right )^{2}\)	\(32\)
default	error in gcdex: invalid arguments\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(3)+14*x^4+18)*ln(x/(ln(3)^2+(-2*x^4+18)*ln(3)+x^8-18*x^4+81))/(x*ln(3)-x^5+9*x),x,method=_RETURNVERB
OSE)

[Out]

ln(x/(ln(3)^2+(-2*x^4+18)*ln(3)+x^8-18*x^4+81))^2

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maxima [B]  time = 0.93, size = 82, normalized size = 4.56 \begin {gather*} -4 \, \log \left (x^{4} - \log \relax (3) - 9\right )^{2} + 4 \, \log \left (x^{4} - \log \relax (3) - 9\right ) \log \relax (x) - \log \relax (x)^{2} - 2 \, {\left (2 \, \log \left (x^{4} - \log \relax (3) - 9\right ) - \log \relax (x)\right )} \log \left (\frac {x}{x^{8} - 18 \, x^{4} - 2 \, {\left (x^{4} - 9\right )} \log \relax (3) + \log \relax (3)^{2} + 81}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(3)+14*x^4+18)*log(x/(log(3)^2+(-2*x^4+18)*log(3)+x^8-18*x^4+81))/(x*log(3)-x^5+9*x),x, algori
thm="maxima")

[Out]

-4*log(x^4 - log(3) - 9)^2 + 4*log(x^4 - log(3) - 9)*log(x) - log(x)^2 - 2*(2*log(x^4 - log(3) - 9) - log(x))*
log(x/(x^8 - 18*x^4 - 2*(x^4 - 9)*log(3) + log(3)^2 + 81))

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mupad [B]  time = 4.18, size = 32, normalized size = 1.78 \begin {gather*} {\ln \left (\frac {x}{{\ln \relax (3)}^2-\ln \relax (3)\,\left (2\,x^4-18\right )-18\,x^4+x^8+81}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x/(log(3)^2 - log(3)*(2*x^4 - 18) - 18*x^4 + x^8 + 81))*(2*log(3) + 14*x^4 + 18))/(9*x + x*log(3) - x
^5),x)

[Out]

log(x/(log(3)^2 - log(3)*(2*x^4 - 18) - 18*x^4 + x^8 + 81))^2

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sympy [A]  time = 0.23, size = 29, normalized size = 1.61 \begin {gather*} \log {\left (\frac {x}{x^{8} - 18 x^{4} + \left (18 - 2 x^{4}\right ) \log {\relax (3 )} + \log {\relax (3 )}^{2} + 81} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(3)+14*x**4+18)*ln(x/(ln(3)**2+(-2*x**4+18)*ln(3)+x**8-18*x**4+81))/(x*ln(3)-x**5+9*x),x)

[Out]

log(x/(x**8 - 18*x**4 + (18 - 2*x**4)*log(3) + log(3)**2 + 81))**2

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