\(\int \frac {(-40 x^2+40 x^3-10 x^4) \log ^2(x)+(2-3 x+x^2) \log (x-x^2)+\log (x) (-2+5 x-2 x^2+(2-2 x+x^2) \log (x-x^2))}{(120 x^2-120 x^3+30 x^4) \log ^2(x)} \, dx\) [2907]

Optimal result

Integrand size = 92, antiderivative size = 37 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=\frac {1}{3} \left (-x+\frac {\log ((1-x) x)}{10 \left (-x+\frac {x}{-1+x}\right ) \log (x)}\right ) \] Output:

1/30*ln((1-x)*x)/(x/(-1+x)-x)/ln(x)-1/3*x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=\frac {1}{30} \left (-10 x-\frac {(-1+x) \log (-((-1+x) x))}{(-2+x) x \log (x)}\right ) \] Input:

Integrate[((-40*x^2 + 40*x^3 - 10*x^4)*Log[x]^2 + (2 - 3*x + x^2)*Log[x - 
x^2] + Log[x]*(-2 + 5*x - 2*x^2 + (2 - 2*x + x^2)*Log[x - x^2]))/((120*x^2 
 - 120*x^3 + 30*x^4)*Log[x]^2),x]
 

Output:

(-10*x - ((-1 + x)*Log[-((-1 + x)*x)])/((-2 + x)*x*Log[x]))/30
 

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.

\(\displaystyle \int \frac {\left (-2 x^2+\left (x^2-2 x+2\right ) \log \left (x-x^2\right )+5 x-2\right ) \log (x)+\left (x^2-3 x+2\right ) \log \left (x-x^2\right )+\left (-10 x^4+40 x^3-40 x^2\right ) \log ^2(x)}{\left (30 x^4-120 x^3+120 x^2\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (-2 x^2+\left (x^2-2 x+2\right ) \log \left (x-x^2\right )+5 x-2\right ) \log (x)+\left (x^2-3 x+2\right ) \log \left (x-x^2\right )+\left (-10 x^4+40 x^3-40 x^2\right ) \log ^2(x)}{x^2 \left (30 x^2-120 x+120\right ) \log ^2(x)}dx\)

\(\Big \downarrow \) 7277

\(\displaystyle 120 \int -\frac {10 \left (x^4-4 x^3+4 x^2\right ) \log ^2(x)+\left (2 x^2-5 x-\left (x^2-2 x+2\right ) \log \left (x-x^2\right )+2\right ) \log (x)-\left (x^2-3 x+2\right ) \log \left (x-x^2\right )}{3600 (2-x)^2 x^2 \log ^2(x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{30} \int \frac {10 \left (x^4-4 x^3+4 x^2\right ) \log ^2(x)+\left (2 x^2-5 x-\left (x^2-2 x+2\right ) \log \left (x-x^2\right )+2\right ) \log (x)-\left (x^2-3 x+2\right ) \log \left (x-x^2\right )}{(2-x)^2 x^2 \log ^2(x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{30} \int \left (\frac {10 \log (x) x^3-20 \log (x) x^2+2 x-1}{(x-2) x^2 \log (x)}+\frac {\left (-\log (x) x^2-x^2+2 \log (x) x+3 x-2 \log (x)-2\right ) \log ((1-x) x)}{(2-x)^2 x^2 \log ^2(x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{30} \left (\frac {1}{2} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)}dx-\int \frac {2 x-1}{(x-2) x^2 \log (x)}dx+\frac {1}{2} \int \frac {\log ((1-x) x)}{x^2 \log (x)}dx-\frac {3}{2} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)}dx-\frac {5}{4} \int \frac {\log ((1-x) x)}{(x-2) \log ^2(x)}dx-\frac {1}{4} \int \frac {\log ((1-x) x)}{x \log ^2(x)}dx+\frac {1}{2} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)}dx-\int \frac {\log ((1-x) x)}{(2-x) \log (x)}dx-\int \frac {\log ((1-x) x)}{(x-2) \log (x)}dx-10 x\right )\)

Input:

Int[((-40*x^2 + 40*x^3 - 10*x^4)*Log[x]^2 + (2 - 3*x + x^2)*Log[x - x^2] + 
 Log[x]*(-2 + 5*x - 2*x^2 + (2 - 2*x + x^2)*Log[x - x^2]))/((120*x^2 - 120 
*x^3 + 30*x^4)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 62.86 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.51

method	result	size
parallelrisch	\(\frac {-60 x^{3} \ln \left (x \right )+80 x^{2} \ln \left (x \right )+80 x \ln \left (x \right )-6 \ln \left (-x^{2}+x \right ) x +6 \ln \left (-x^{2}+x \right )}{180 x \ln \left (x \right ) \left (-2+x \right )}\)	\(56\)
risch	\(-\frac {\left (-1+x \right ) \ln \left (-1+x \right )}{30 x \left (-2+x \right ) \ln \left (x \right )}-\frac {2 i \pi x -i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+2 i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2}-2 i \pi -i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{3}-i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-1+x \right )\right )-2 i \pi x \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2}+i \pi x \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{3}-i \pi x \,\operatorname {csgn}\left (i x \left (-1+x \right )\right ) \operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (i x \right )+i \pi x \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-1+x \right )\right )+i \pi \,\operatorname {csgn}\left (i x \left (-1+x \right )\right ) \operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (i x \right )+i \pi x \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+20 x^{3} \ln \left (x \right )-40 x^{2} \ln \left (x \right )+2 x \ln \left (x \right )-2 \ln \left (x \right )}{60 x \left (-2+x \right ) \ln \left (x \right )}\)	\(259\)

Input:

int(((-10*x^4+40*x^3-40*x^2)*ln(x)^2+((x^2-2*x+2)*ln(-x^2+x)-2*x^2+5*x-2)* 
ln(x)+(x^2-3*x+2)*ln(-x^2+x))/(30*x^4-120*x^3+120*x^2)/ln(x)^2,x,method=_R 
ETURNVERBOSE)
 

Output:

1/180*(-60*x^3*ln(x)+80*x^2*ln(x)+80*x*ln(x)-6*ln(-x^2+x)*x+6*ln(-x^2+x))/ 
x/ln(x)/(-2+x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=-\frac {{\left (x - 1\right )} \log \left (-x^{2} + x\right ) + 10 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right )}{30 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right )} \] Input:

integrate(((-10*x^4+40*x^3-40*x^2)*log(x)^2+((x^2-2*x+2)*log(-x^2+x)-2*x^2 
+5*x-2)*log(x)+(x^2-3*x+2)*log(-x^2+x))/(30*x^4-120*x^3+120*x^2)/log(x)^2, 
x, algorithm="fricas")
 

Output:

-1/30*((x - 1)*log(-x^2 + x) + 10*(x^3 - 2*x^2)*log(x))/((x^2 - 2*x)*log(x 
))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((-10*x**4+40*x**3-40*x**2)*ln(x)**2+((x**2-2*x+2)*ln(-x**2+x)-2 
*x**2+5*x-2)*ln(x)+(x**2-3*x+2)*ln(-x**2+x))/(30*x**4-120*x**3+120*x**2)/l 
n(x)**2,x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=-\frac {{\left (10 \, x^{3} - 20 \, x^{2} + x - 1\right )} \log \left (x\right ) + {\left (x - 1\right )} \log \left (-x + 1\right )}{30 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right )} \] Input:

integrate(((-10*x^4+40*x^3-40*x^2)*log(x)^2+((x^2-2*x+2)*log(-x^2+x)-2*x^2 
+5*x-2)*log(x)+(x^2-3*x+2)*log(-x^2+x))/(30*x^4-120*x^3+120*x^2)/log(x)^2, 
x, algorithm="maxima")
 

Output:

-1/30*((10*x^3 - 20*x^2 + x - 1)*log(x) + (x - 1)*log(-x + 1))/((x^2 - 2*x 
)*log(x))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=-\frac {1}{3} \, x - \frac {{\left (x - 1\right )} \log \left (-x + 1\right )}{30 \, {\left (x^{2} \log \left (x\right ) - 2 \, x \log \left (x\right )\right )}} - \frac {1}{60 \, {\left (x - 2\right )}} - \frac {1}{60 \, x} \] Input:

integrate(((-10*x^4+40*x^3-40*x^2)*log(x)^2+((x^2-2*x+2)*log(-x^2+x)-2*x^2 
+5*x-2)*log(x)+(x^2-3*x+2)*log(-x^2+x))/(30*x^4-120*x^3+120*x^2)/log(x)^2, 
x, algorithm="giac")
 

Output:

-1/3*x - 1/30*(x - 1)*log(-x + 1)/(x^2*log(x) - 2*x*log(x)) - 1/60/(x - 2) 
 - 1/60/x
 

Mupad [B] (verification not implemented)

Time = 2.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=\frac {\frac {\ln \left (x-x^2\right )}{30}-\frac {x\,\ln \left (x-x^2\right )}{30}}{x\,\ln \left (x\right )\,\left (x-2\right )}-\frac {x}{3} \] Input:

int((log(x)*(5*x - 2*x^2 + log(x - x^2)*(x^2 - 2*x + 2) - 2) - log(x)^2*(4 
0*x^2 - 40*x^3 + 10*x^4) + log(x - x^2)*(x^2 - 3*x + 2))/(log(x)^2*(120*x^ 
2 - 120*x^3 + 30*x^4)),x)
 

Output:

(log(x - x^2)/30 - (x*log(x - x^2))/30)/(x*log(x)*(x - 2)) - x/3
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.30 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=\frac {-\mathrm {log}\left (-x^{2}+x \right ) x +\mathrm {log}\left (-x^{2}+x \right )-10 \,\mathrm {log}\left (x \right ) x^{3}+20 \,\mathrm {log}\left (x \right ) x^{2}}{30 \,\mathrm {log}\left (x \right ) x \left (x -2\right )} \] Input:

int(((-10*x^4+40*x^3-40*x^2)*log(x)^2+((x^2-2*x+2)*log(-x^2+x)-2*x^2+5*x-2 
)*log(x)+(x^2-3*x+2)*log(-x^2+x))/(30*x^4-120*x^3+120*x^2)/log(x)^2,x)
 

Output:

( - log( - x**2 + x)*x + log( - x**2 + x) - 10*log(x)*x**3 + 20*log(x)*x** 
2)/(30*log(x)*x*(x - 2))