\(\int \frac {(-18-6 x+9 x^2+3 x^3) \log ^2(x) \log (2 x-x^3)+\log ^3(x) (12+4 x-18 x^2-6 x^3+(30+8 x-15 x^2-4 x^3) \log (2 x-x^3))}{(-2 x^6+x^8) \log ^3(2 x-x^3)} \, dx\) [2780]

Optimal result

Integrand size = 101, antiderivative size = 23 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {(3+x) \log ^3(x)}{x^5 \log ^2\left (x \left (2-x^2\right )\right )} \] Output:

ln(x)^3/x^5/ln(x*(-x^2+2))^2*(3+x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {(3+x) \log ^3(x)}{x^5 \log ^2\left (-x \left (-2+x^2\right )\right )} \] Input:

Integrate[((-18 - 6*x + 9*x^2 + 3*x^3)*Log[x]^2*Log[2*x - x^3] + Log[x]^3* 
(12 + 4*x - 18*x^2 - 6*x^3 + (30 + 8*x - 15*x^2 - 4*x^3)*Log[2*x - x^3]))/ 
((-2*x^6 + x^8)*Log[2*x - x^3]^3),x]
 

Output:

((3 + x)*Log[x]^3)/(x^5*Log[-(x*(-2 + x^2))]^2)
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (-6 x^3-18 x^2+\left (-4 x^3-15 x^2+8 x+30\right ) \log \left (2 x-x^3\right )+4 x+12\right ) \log ^3(x)+\left (3 x^3+9 x^2-6 x-18\right ) \log \left (2 x-x^3\right ) \log ^2(x)}{\left (x^8-2 x^6\right ) \log ^3\left (2 x-x^3\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (-6 x^3-18 x^2+\left (-4 x^3-15 x^2+8 x+30\right ) \log \left (2 x-x^3\right )+4 x+12\right ) \log ^3(x)+\left (3 x^3+9 x^2-6 x-18\right ) \log \left (2 x-x^3\right ) \log ^2(x)}{x^6 \left (x^2-2\right ) \log ^3\left (2 x-x^3\right )}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {(3 x-4 x \log (x)-15 \log (x)+9) \log ^2(x)}{x^6 \log ^2\left (x \left (2-x^2\right )\right )}+\frac {2 \left (3 x^3+9 x^2-2 x-6\right ) \log ^3(x)}{x^6 \left (2-x^2\right ) \log ^3\left (x \left (2-x^2\right )\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \int \frac {\log ^3(x)}{\left (\sqrt {2}-x\right ) \log ^3\left (x \left (2-x^2\right )\right )}dx}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\log ^3(x)}{\left (\sqrt {2}-x\right ) \log ^3\left (x \left (2-x^2\right )\right )}dx+3 \int \frac {\log ^3(x)}{x^2 \log ^3\left (x \left (2-x^2\right )\right )}dx+\int \frac {\log ^3(x)}{x \log ^3\left (x \left (2-x^2\right )\right )}dx+\frac {3 \int \frac {\log ^3(x)}{\left (x+\sqrt {2}\right ) \log ^3\left (x \left (2-x^2\right )\right )}dx}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\log ^3(x)}{\left (x+\sqrt {2}\right ) \log ^3\left (x \left (2-x^2\right )\right )}dx-6 \int \frac {\log ^3(x)}{x^6 \log ^3\left (x \left (2-x^2\right )\right )}dx+9 \int \frac {\log ^2(x)}{x^6 \log ^2\left (x \left (2-x^2\right )\right )}dx-15 \int \frac {\log ^3(x)}{x^6 \log ^2\left (x \left (2-x^2\right )\right )}dx-2 \int \frac {\log ^3(x)}{x^5 \log ^3\left (x \left (2-x^2\right )\right )}dx+3 \int \frac {\log ^2(x)}{x^5 \log ^2\left (x \left (2-x^2\right )\right )}dx-4 \int \frac {\log ^3(x)}{x^5 \log ^2\left (x \left (2-x^2\right )\right )}dx+6 \int \frac {\log ^3(x)}{x^4 \log ^3\left (x \left (2-x^2\right )\right )}dx+2 \int \frac {\log ^3(x)}{x^3 \log ^3\left (x \left (2-x^2\right )\right )}dx\)

Input:

Int[((-18 - 6*x + 9*x^2 + 3*x^3)*Log[x]^2*Log[2*x - x^3] + Log[x]^3*(12 + 
4*x - 18*x^2 - 6*x^3 + (30 + 8*x - 15*x^2 - 4*x^3)*Log[2*x - x^3]))/((-2*x 
^6 + x^8)*Log[2*x - x^3]^3),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.30 (sec) , antiderivative size = 835, normalized size of antiderivative = 36.30

\[\text {Expression too large to display}\]

Input:

int((((-4*x^3-15*x^2+8*x+30)*ln(-x^3+2*x)-6*x^3-18*x^2+4*x+12)*ln(x)^3+(3* 
x^3+9*x^2-6*x-18)*ln(-x^3+2*x)*ln(x)^2)/(x^8-2*x^6)/ln(-x^3+2*x)^3,x)
 

Output:

2*ln(x)^2*(-36*ln(x^2-2)+12*x^3*ln(x)-16*x*ln(x)+36*x^2*ln(x)-48*ln(x)+18* 
I*Pi*x^2+6*I*Pi*x^3-36*I*Pi-18*I*Pi*csgn(I*x*(x^2-2))^3+36*I*Pi*csgn(I*x*( 
x^2-2))^2-12*I*x*Pi-12*x*ln(x^2-2)+6*x^3*ln(x^2-2)+18*x^2*ln(x^2-2)-9*I*x^ 
2*Pi*csgn(I*(x^2-2))*csgn(I*x)*csgn(I*x*(x^2-2))-3*I*x^3*Pi*csgn(I*(x^2-2) 
)*csgn(I*x)*csgn(I*x*(x^2-2))+6*I*x*Pi*csgn(I*(x^2-2))*csgn(I*x)*csgn(I*x* 
(x^2-2))-6*I*x*Pi*csgn(I*x*(x^2-2))^3+9*I*x^2*Pi*csgn(I*x*(x^2-2))^3+3*I*x 
^3*Pi*csgn(I*x*(x^2-2))^3-18*I*Pi*csgn(I*(x^2-2))*csgn(I*x*(x^2-2))^2-18*I 
*Pi*csgn(I*x)*csgn(I*x*(x^2-2))^2-18*I*x^2*Pi*csgn(I*x*(x^2-2))^2-6*I*x^3* 
Pi*csgn(I*x*(x^2-2))^2+12*I*x*Pi*csgn(I*x*(x^2-2))^2-6*I*x*Pi*csgn(I*x)*cs 
gn(I*x*(x^2-2))^2+3*I*x^3*Pi*csgn(I*(x^2-2))*csgn(I*x*(x^2-2))^2+3*I*x^3*P 
i*csgn(I*x)*csgn(I*x*(x^2-2))^2+9*I*x^2*Pi*csgn(I*(x^2-2))*csgn(I*x*(x^2-2 
))^2+18*I*Pi*csgn(I*(x^2-2))*csgn(I*x)*csgn(I*x*(x^2-2))+9*I*x^2*Pi*csgn(I 
*x)*csgn(I*x*(x^2-2))^2-6*I*x*Pi*csgn(I*(x^2-2))*csgn(I*x*(x^2-2))^2)/(2*I 
*Pi+2*ln(x)+2*ln(x^2-2)-I*Pi*csgn(I*(x^2-2))*csgn(I*x)*csgn(I*x*(x^2-2))+I 
*Pi*csgn(I*(x^2-2))*csgn(I*x*(x^2-2))^2+I*Pi*csgn(I*x)*csgn(I*x*(x^2-2))^2 
-2*I*Pi*csgn(I*x*(x^2-2))^2+I*Pi*csgn(I*x*(x^2-2))^3)^2/(3*x^2-2)/x^5+6*I* 
(x^2-2)*(3+x)*ln(x)^2/(3*x^2-2)/x^5/(-2*Pi*csgn(I*x*(x^2-2))^2-Pi*csgn(I*( 
x^2-2))*csgn(I*x)*csgn(I*x*(x^2-2))+Pi*csgn(I*(x^2-2))*csgn(I*x*(x^2-2))^2 
+Pi*csgn(I*x)*csgn(I*x*(x^2-2))^2+Pi*csgn(I*x*(x^2-2))^3+2*Pi-2*I*ln(x^2-2 
)-2*I*ln(x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {{\left (x + 3\right )} \log \left (x\right )^{3}}{x^{5} \log \left (-x^{3} + 2 \, x\right )^{2}} \] Input:

integrate((((-4*x^3-15*x^2+8*x+30)*log(-x^3+2*x)-6*x^3-18*x^2+4*x+12)*log( 
x)^3+(3*x^3+9*x^2-6*x-18)*log(-x^3+2*x)*log(x)^2)/(x^8-2*x^6)/log(-x^3+2*x 
)^3,x, algorithm="fricas")
 

Output:

(x + 3)*log(x)^3/(x^5*log(-x^3 + 2*x)^2)
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {x \log {\left (x \right )}^{3} + 3 \log {\left (x \right )}^{3}}{x^{5} \log {\left (- x^{3} + 2 x \right )}^{2}} \] Input:

integrate((((-4*x**3-15*x**2+8*x+30)*ln(-x**3+2*x)-6*x**3-18*x**2+4*x+12)* 
ln(x)**3+(3*x**3+9*x**2-6*x-18)*ln(-x**3+2*x)*ln(x)**2)/(x**8-2*x**6)/ln(- 
x**3+2*x)**3,x)
 

Output:

(x*log(x)**3 + 3*log(x)**3)/(x**5*log(-x**3 + 2*x)**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).

Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {{\left (x + 3\right )} \log \left (x\right )^{3}}{x^{5} \log \left (-x^{2} + 2\right )^{2} + 2 \, x^{5} \log \left (-x^{2} + 2\right ) \log \left (x\right ) + x^{5} \log \left (x\right )^{2}} \] Input:

integrate((((-4*x^3-15*x^2+8*x+30)*log(-x^3+2*x)-6*x^3-18*x^2+4*x+12)*log( 
x)^3+(3*x^3+9*x^2-6*x-18)*log(-x^3+2*x)*log(x)^2)/(x^8-2*x^6)/log(-x^3+2*x 
)^3,x, algorithm="maxima")
 

Output:

(x + 3)*log(x)^3/(x^5*log(-x^2 + 2)^2 + 2*x^5*log(-x^2 + 2)*log(x) + x^5*l 
og(x)^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (22) = 44\).

Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.96 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {3 \, x^{3} \log \left (x\right )^{3} + 9 \, x^{2} \log \left (x\right )^{3} - 2 \, x \log \left (x\right )^{3} - 6 \, \log \left (x\right )^{3}}{3 \, x^{7} \log \left (-x^{2} + 2\right )^{2} + 6 \, x^{7} \log \left (-x^{2} + 2\right ) \log \left (x\right ) + 3 \, x^{7} \log \left (x\right )^{2} - 2 \, x^{5} \log \left (-x^{2} + 2\right )^{2} - 4 \, x^{5} \log \left (-x^{2} + 2\right ) \log \left (x\right ) - 2 \, x^{5} \log \left (x\right )^{2}} \] Input:

integrate((((-4*x^3-15*x^2+8*x+30)*log(-x^3+2*x)-6*x^3-18*x^2+4*x+12)*log( 
x)^3+(3*x^3+9*x^2-6*x-18)*log(-x^3+2*x)*log(x)^2)/(x^8-2*x^6)/log(-x^3+2*x 
)^3,x, algorithm="giac")
 

Output:

(3*x^3*log(x)^3 + 9*x^2*log(x)^3 - 2*x*log(x)^3 - 6*log(x)^3)/(3*x^7*log(- 
x^2 + 2)^2 + 6*x^7*log(-x^2 + 2)*log(x) + 3*x^7*log(x)^2 - 2*x^5*log(-x^2 
+ 2)^2 - 4*x^5*log(-x^2 + 2)*log(x) - 2*x^5*log(x)^2)
 

Mupad [B] (verification not implemented)

Time = 2.97 (sec) , antiderivative size = 467, normalized size of antiderivative = 20.30 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {\frac {{\ln \left (x\right )}^3\,\left (x+3\right )}{x^5}-\frac {\ln \left (2\,x-x^3\right )\,{\ln \left (x\right )}^2\,\left (x^2-2\right )\,\left (3\,x-15\,\ln \left (x\right )-4\,x\,\ln \left (x\right )+9\right )}{2\,x^5\,\left (3\,x^2-2\right )}}{{\ln \left (2\,x-x^3\right )}^2}+\frac {\frac {\left (x^2-2\right )\,\left (3\,x\,{\ln \left (x\right )}^2-4\,x\,{\ln \left (x\right )}^3+9\,{\ln \left (x\right )}^2-15\,{\ln \left (x\right )}^3\right )}{2\,x^5\,\left (3\,x^2-2\right )}-\frac {\ln \left (2\,x-x^3\right )\,\left (x^2-2\right )\,\left (48\,x^5\,{\ln \left (x\right )}^3-72\,x^5\,{\ln \left (x\right )}^2+18\,x^5\,\ln \left (x\right )+225\,x^4\,{\ln \left (x\right )}^3-270\,x^4\,{\ln \left (x\right )}^2+54\,x^4\,\ln \left (x\right )-160\,x^3\,{\ln \left (x\right )}^3+216\,x^3\,{\ln \left (x\right )}^2-48\,x^3\,\ln \left (x\right )-720\,x^2\,{\ln \left (x\right )}^3+792\,x^2\,{\ln \left (x\right )}^2-144\,x^2\,\ln \left (x\right )+64\,x\,{\ln \left (x\right )}^3-96\,x\,{\ln \left (x\right )}^2+24\,x\,\ln \left (x\right )+300\,{\ln \left (x\right )}^3-360\,{\ln \left (x\right )}^2+72\,\ln \left (x\right )\right )}{2\,x^5\,{\left (3\,x^2-2\right )}^3}}{\ln \left (2\,x-x^3\right )}-\frac {{\ln \left (x\right )}^2\,\left (-\frac {4\,x^7}{3}-5\,x^6+\frac {20\,x^5}{3}+\frac {74\,x^4}{3}-\frac {88\,x^3}{9}-36\,x^2+\frac {32\,x}{9}+\frac {40}{3}\right )}{-x^{11}+2\,x^9-\frac {4\,x^7}{3}+\frac {8\,x^5}{27}}+\frac {{\ln \left (x\right )}^3\,\left (-\frac {8\,x^7}{9}-\frac {25\,x^6}{6}+\frac {128\,x^5}{27}+\frac {65\,x^4}{3}-\frac {64\,x^3}{9}-\frac {290\,x^2}{9}+\frac {64\,x}{27}+\frac {100}{9}\right )}{-x^{11}+2\,x^9-\frac {4\,x^7}{3}+\frac {8\,x^5}{27}}+\frac {\ln \left (x\right )\,\left (\frac {x^5}{3}+x^4-\frac {4\,x^3}{3}-4\,x^2+\frac {4\,x}{3}+4\right )}{x^9-\frac {4\,x^7}{3}+\frac {4\,x^5}{9}} \] Input:

int(-(log(x)^3*(4*x - 18*x^2 - 6*x^3 + log(2*x - x^3)*(8*x - 15*x^2 - 4*x^ 
3 + 30) + 12) - log(2*x - x^3)*log(x)^2*(6*x - 9*x^2 - 3*x^3 + 18))/(log(2 
*x - x^3)^3*(2*x^6 - x^8)),x)
 

Output:

((log(x)^3*(x + 3))/x^5 - (log(2*x - x^3)*log(x)^2*(x^2 - 2)*(3*x - 15*log 
(x) - 4*x*log(x) + 9))/(2*x^5*(3*x^2 - 2)))/log(2*x - x^3)^2 + (((x^2 - 2) 
*(3*x*log(x)^2 - 4*x*log(x)^3 + 9*log(x)^2 - 15*log(x)^3))/(2*x^5*(3*x^2 - 
 2)) - (log(2*x - x^3)*(x^2 - 2)*(72*log(x) - 96*x*log(x)^2 - 144*x^2*log( 
x) + 64*x*log(x)^3 - 48*x^3*log(x) + 54*x^4*log(x) + 18*x^5*log(x) - 360*l 
og(x)^2 + 300*log(x)^3 + 792*x^2*log(x)^2 - 720*x^2*log(x)^3 + 216*x^3*log 
(x)^2 - 160*x^3*log(x)^3 - 270*x^4*log(x)^2 + 225*x^4*log(x)^3 - 72*x^5*lo 
g(x)^2 + 48*x^5*log(x)^3 + 24*x*log(x)))/(2*x^5*(3*x^2 - 2)^3))/log(2*x - 
x^3) - (log(x)^2*((32*x)/9 - 36*x^2 - (88*x^3)/9 + (74*x^4)/3 + (20*x^5)/3 
 - 5*x^6 - (4*x^7)/3 + 40/3))/((8*x^5)/27 - (4*x^7)/3 + 2*x^9 - x^11) + (l 
og(x)^3*((64*x)/27 - (290*x^2)/9 - (64*x^3)/9 + (65*x^4)/3 + (128*x^5)/27 
- (25*x^6)/6 - (8*x^7)/9 + 100/9))/((8*x^5)/27 - (4*x^7)/3 + 2*x^9 - x^11) 
 + (log(x)*((4*x)/3 - 4*x^2 - (4*x^3)/3 + x^4 + x^5/3 + 4))/((4*x^5)/9 - ( 
4*x^7)/3 + x^9)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-18-6 x+9 x^2+3 x^3\right ) \log ^2(x) \log \left (2 x-x^3\right )+\log ^3(x) \left (12+4 x-18 x^2-6 x^3+\left (30+8 x-15 x^2-4 x^3\right ) \log \left (2 x-x^3\right )\right )}{\left (-2 x^6+x^8\right ) \log ^3\left (2 x-x^3\right )} \, dx=\frac {\mathrm {log}\left (x \right )^{3} \left (x +3\right )}{\mathrm {log}\left (-x^{3}+2 x \right )^{2} x^{5}} \] Input:

int((((-4*x^3-15*x^2+8*x+30)*log(-x^3+2*x)-6*x^3-18*x^2+4*x+12)*log(x)^3+( 
3*x^3+9*x^2-6*x-18)*log(-x^3+2*x)*log(x)^2)/(x^8-2*x^6)/log(-x^3+2*x)^3,x)
 

Output:

(log(x)**3*(x + 3))/(log( - x**3 + 2*x)**2*x**5)