\(\int \frac {(18+14 x^4+2 \log (3)) \log (\frac {x}{81-18 x^4+x^8+(18-2 x^4) \log (3)+\log ^2(3)})}{9 x-x^5+x \log (3)} \, dx\) [1880]

Optimal result

Integrand size = 56, antiderivative size = 18 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx=4+\log ^2\left (\frac {x}{\left (-9+x^4-\log (3)\right )^2}\right ) \] Output:

ln(x/(x^4-9-ln(3))^2)^2+4
 

Mathematica [C] (warning: unable to verify)

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

Time = 0.40 (sec) , antiderivative size = 1485, normalized size of antiderivative = 82.50 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx =\text {Too large to display} \] Input:

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]
 

Output:

-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*L 
og[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*L 
og[((-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 + Log[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 + L 
og[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 + 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 + Lo 
g[3])^(1/4)))/(9 + Log[3])^(1/4)] - (Log[9 + Log[3]]*Log[-x + (9 + Log[...
 

Rubi [C] (verified)

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

Time = 3.86 (sec) , antiderivative size = 1299, normalized size of antiderivative = 72.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6, 2026, 3008, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6

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

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (14 x^4+18+2 \log (3)\right ) \log \left (\frac {x}{x^8-18 x^4+\left (18-2 x^4\right ) \log (3)+81+\log ^2(3)}\right )}{x \left (-x^4+9+\log (3)\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle \int \left (\frac {2 \log \left (\frac {x}{x^8-18 x^4+\left (18-2 x^4\right ) \log (3)+81+\log ^2(3)}\right )}{x}-\frac {16 x^3 \log \left (\frac {x}{x^8-18 x^4+\left (18-2 x^4\right ) \log (3)+81+\log ^2(3)}\right )}{x^4-9-\log (3)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\log ^2(x)+2 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log (x)+4 \log \left (1-\frac {x^4}{9+\log (3)}\right ) \log (x)+2 \log (9+\log (3)) \log (x)-4 \log ^2\left (i \sqrt [4]{9+\log (3)}-x\right )-4 \log ^2\left (x+i \sqrt [4]{9+\log (3)}\right )-4 \log ^2\left (\sqrt [4]{9+\log (3)}-x\right )-4 \log ^2\left (x+\sqrt [4]{9+\log (3)}\right )+4 \log \left (-\frac {i x}{\sqrt [4]{9+\log (3)}}\right ) \log \left (i \sqrt [4]{9+\log (3)}-x\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (i \sqrt [4]{9+\log (3)}-x\right )-8 \log \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x-\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (i \sqrt [4]{9+\log (3)}-x\right )+4 \log \left (\frac {i x}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x-\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-8 \log \left (\frac {i \left (x-i \sqrt [4]{9+\log (3)}\right )}{2 \sqrt [4]{9+\log (3)}}\right ) \log \left (x+i \sqrt [4]{9+\log (3)}\right )-8 \log \left (i \sqrt [4]{9+\log (3)}-x\right ) \log \left (-\frac {i \left (x+i \sqrt [4]{9+\log (3)}\right )}{2 \sqrt [4]{9+\log (3)}}\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (\sqrt [4]{9+\log (3)}-x\right )-8 \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x-i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (\sqrt [4]{9+\log (3)}-x\right )-8 \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x+i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (\sqrt [4]{9+\log (3)}-x\right )-4 \log \left (\frac {x}{x^8-18 x^4+\log ^2(3)+2 \left (9-x^4\right ) \log (3)+81}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {x-\sqrt [4]{9+\log (3)}}{2 \sqrt [4]{9+\log (3)}}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x-i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x+i \sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right ) \log \left (x+\sqrt [4]{9+\log (3)}\right )-8 \log \left (\sqrt [4]{9+\log (3)}-x\right ) \log \left (\frac {x+\sqrt [4]{9+\log (3)}}{2 \sqrt [4]{9+\log (3)}}\right )-8 \log \left (i \sqrt [4]{9+\log (3)}-x\right ) \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x+\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right )-8 \log \left (x+i \sqrt [4]{9+\log (3)}\right ) \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (x+\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right )+\operatorname {PolyLog}\left (2,\frac {x^4}{9+\log (3)}\right )-4 \operatorname {PolyLog}\left (2,-\frac {x}{\sqrt [4]{9+\log (3)}}\right )-4 \operatorname {PolyLog}\left (2,\frac {x}{\sqrt [4]{9+\log (3)}}\right )-8 \operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1-\frac {x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\frac {i x}{\sqrt [4]{9+\log (3)}}\right )\right )-8 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1-\frac {i x}{\sqrt [4]{9+\log (3)}}\right )\right )+4 \operatorname {PolyLog}\left (2,1-\frac {i x}{\sqrt [4]{9+\log (3)}}\right )-8 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (\frac {i x}{\sqrt [4]{9+\log (3)}}+1\right )\right )-8 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {i x}{\sqrt [4]{9+\log (3)}}+1\right )\right )+4 \operatorname {PolyLog}\left (2,\frac {i x}{\sqrt [4]{9+\log (3)}}+1\right )-8 \operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {x}{\sqrt [4]{9+\log (3)}}+i\right )\right )-8 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (\frac {x}{\sqrt [4]{9+\log (3)}}+1\right )\right )-8 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {x}{\sqrt [4]{9+\log (3)}}+1\right )\right )-8 \operatorname {PolyLog}\left (2,\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (x+\sqrt [4]{9+\log (3)}\right )}{\sqrt [4]{9+\log (3)}}\right )\)

Input:

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]
 

Output:

-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/(8 
1 - 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)] - 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 + L 
og[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 + Log[3])^(1/4 
)] - 8*Log[((1/2 - I/2)*(x + I*(9 + Log[3])^(1/4)))/(9 + Log[3])^(1/4)]*Lo 
g[-x + (9 + Log[3])^(1/4)] - 4*Log[-x + (9 + Log[3])^(1/4)]^2 - 4*Log[x/(8 
1 - 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 +...
 

Defintions of rubi rules used

rule 6

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 3008

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] && RationalFuncti 
onQ[RGx, x] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78

method	result	size
norman	\(\ln \left (\frac {x}{\ln \left (3\right )^{2}+\left (-2 x^{4}+18\right ) \ln \left (3\right )+x^{8}-18 x^{4}+81}\right )^{2}\)	\(32\)
risch	\(\ln \left (\frac {x}{\ln \left (3\right )^{2}+\left (-2 x^{4}+18\right ) \ln \left (3\right )+x^{8}-18 x^{4}+81}\right )^{2}\)	\(32\)
default	\(\ln \left (\frac {x}{x^{8}-2 x^{4} \ln \left (3\right )-18 x^{4}+\ln \left (3\right )^{2}+18 \ln \left (3\right )+81}\right )^{2}\)	\(33\)

Input:

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=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx=\log \left (\frac {x}{x^{8} - 18 \, x^{4} - 2 \, {\left (x^{4} - 9\right )} \log \left (3\right ) + \log \left (3\right )^{2} + 81}\right )^{2} \] Input:

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, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx=\log {\left (\frac {x}{x^{8} - 18 x^{4} + \left (18 - 2 x^{4}\right ) \log {\left (3 \right )} + \log {\left (3 \right )}^{2} + 81} \right )}^{2} \] Input:

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)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (18) = 36\).

Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.56 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx=-4 \, \log \left (x^{4} - \log \left (3\right ) - 9\right )^{2} + 4 \, \log \left (x^{4} - \log \left (3\right ) - 9\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, {\left (2 \, \log \left (x^{4} - \log \left (3\right ) - 9\right ) - \log \left (x\right )\right )} \log \left (\frac {x}{x^{8} - 18 \, x^{4} - 2 \, {\left (x^{4} - 9\right )} \log \left (3\right ) + \log \left (3\right )^{2} + 81}\right ) \] Input:

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, algorithm="maxima")
 

Output:

-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))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (18) = 36\).

Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.56 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx=2 \, {\left (2 \, \log \left (x^{4} - \log \left (3\right ) - 9\right ) - \log \left (x\right )\right )} \log \left (x^{8} - 2 \, x^{4} \log \left (3\right ) - 18 \, x^{4} + \log \left (3\right )^{2} + 18 \, \log \left (3\right ) + 81\right ) - 4 \, \log \left (x^{4} - \log \left (3\right ) - 9\right )^{2} + \log \left (x\right )^{2} \] Input:

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, algorithm="giac")
 

Output:

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
 

Mupad [B] (verification not implemented)

Time = 5.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx={\ln \left (\frac {x}{{\ln \left (3\right )}^2-\ln \left (3\right )\,\left (2\,x^4-18\right )-18\,x^4+x^8+81}\right )}^2 \] Input:

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)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {\left (18+14 x^4+2 \log (3)\right ) \log \left (\frac {x}{81-18 x^4+x^8+\left (18-2 x^4\right ) \log (3)+\log ^2(3)}\right )}{9 x-x^5+x \log (3)} \, dx=\mathrm {log}\left (\frac {x}{\mathrm {log}\left (3\right )^{2}-2 \,\mathrm {log}\left (3\right ) x^{4}+18 \,\mathrm {log}\left (3\right )+x^{8}-18 x^{4}+81}\right )^{2} \] Input:

int((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)
 

Output:

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