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

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

-1/3*x + Defer[Int][(1 - 2*x)/((-2 + x)*x^2*Log[x]), x]/30 + Defer[Int][Log[(1 - x)*x]/((2 - x)*Log[x]^2), x]/
20 + (7*Defer[Int][Log[(1 - x)*x]/((-2 + x)*Log[x]^2), x])/120 + Defer[Int][Log[(1 - x)*x]/(x^2*Log[x]^2), x]/
60 - Defer[Int][Log[(1 - x)*x]/(x*Log[x]^2), x]/120 + Defer[Int][Log[(1 - x)*x]/((2 - x)^2*Log[x]), x]/60 + De
fer[Int][Log[(1 - x)*x]/((2 - x)*Log[x]), x]/20 + Defer[Int][Log[(1 - x)*x]/((-2 + x)*Log[x]), x]/20 + Defer[I
nt][Log[(1 - x)*x]/(x^2*Log[x]), x]/60

Rubi steps \begin{align*} \text {integral}& = \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 )}{x^2 \left (120-120 x+30 x^2\right ) \log ^2(x)} \, dx \\ & = \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 )}{30 (-2+x)^2 x^2 \log ^2(x)} \, dx \\ & = \frac {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 )}{(-2+x)^2 x^2 \log ^2(x)} \, dx \\ & = \frac {1}{30} \int \left (\frac {1-2 x+20 x^2 \log (x)-10 x^3 \log (x)}{(-2+x) x^2 \log (x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x)^2 x^2 \log ^2(x)}\right ) \, dx \\ & = \frac {1}{30} \int \frac {1-2 x+20 x^2 \log (x)-10 x^3 \log (x)}{(-2+x) x^2 \log (x)} \, dx+\frac {1}{30} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x)^2 x^2 \log ^2(x)} \, dx \\ & = \frac {1}{30} \int \left (-10+\frac {1-2 x}{(-2+x) x^2 \log (x)}\right ) \, dx+\frac {1}{30} \int \left (\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 (2-x)^2 \log ^2(x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 (2-x) \log ^2(x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 x^2 \log ^2(x)}+\frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{4 x \log ^2(x)}\right ) \, dx \\ & = -\frac {x}{3}+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\left (2-3 x+x^2+2 \log (x)-2 x \log (x)+x^2 \log (x)\right ) \log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx \\ & = -\frac {x}{3}+\frac {1}{120} \int \left (\frac {\log ((1-x) x)}{\log ^2(x)}+\frac {2 \log ((1-x) x)}{x^2 \log ^2(x)}-\frac {3 \log ((1-x) x)}{x \log ^2(x)}+\frac {\log ((1-x) x)}{\log (x)}+\frac {2 \log ((1-x) x)}{x^2 \log (x)}-\frac {2 \log ((1-x) x)}{x \log (x)}\right ) \, dx+\frac {1}{120} \int \left (-\frac {3 \log ((1-x) x)}{\log ^2(x)}+\frac {2 \log ((1-x) x)}{x \log ^2(x)}+\frac {x \log ((1-x) x)}{\log ^2(x)}-\frac {2 \log ((1-x) x)}{\log (x)}+\frac {2 \log ((1-x) x)}{x \log (x)}+\frac {x \log ((1-x) x)}{\log (x)}\right ) \, dx+\frac {1}{120} \int \left (\frac {2 \log ((1-x) x)}{(2-x)^2 \log ^2(x)}-\frac {3 x \log ((1-x) x)}{(2-x)^2 \log ^2(x)}+\frac {x^2 \log ((1-x) x)}{(2-x)^2 \log ^2(x)}+\frac {2 \log ((1-x) x)}{(2-x)^2 \log (x)}-\frac {2 x \log ((1-x) x)}{(2-x)^2 \log (x)}+\frac {x^2 \log ((1-x) x)}{(2-x)^2 \log (x)}\right ) \, dx+\frac {1}{120} \int \left (\frac {2 \log ((1-x) x)}{(2-x) \log ^2(x)}+\frac {3 x \log ((1-x) x)}{(-2+x) \log ^2(x)}+\frac {x^2 \log ((1-x) x)}{(2-x) \log ^2(x)}+\frac {2 \log ((1-x) x)}{(2-x) \log (x)}+\frac {2 x \log ((1-x) x)}{(-2+x) \log (x)}+\frac {x^2 \log ((1-x) x)}{(2-x) \log (x)}\right ) \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx \\ & = -\frac {x}{3}+\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{120} \int \frac {x \log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{120} \int \frac {\log ((1-x) x)}{\log (x)} \, dx+\frac {1}{120} \int \frac {x \log ((1-x) x)}{\log (x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{120} \int \frac {x^2 \log ((1-x) x)}{(2-x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{\log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx-\frac {1}{60} \int \frac {x \log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {x \log ((1-x) x)}{(-2+x) \log (x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx-\frac {1}{40} \int \frac {x \log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{40} \int \frac {x \log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx \\ & = -\frac {x}{3}-\frac {1}{120} \log ((1-x) x) \text {li}(x)+\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{120} \int \frac {x \log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{120} \int \frac {x \log ((1-x) x)}{\log (x)} \, dx+\frac {1}{120} \int \left (\frac {\log ((1-x) x)}{\log ^2(x)}+\frac {4 \log ((1-x) x)}{(2-x)^2 \log ^2(x)}+\frac {4 \log ((1-x) x)}{(-2+x) \log ^2(x)}\right ) \, dx+\frac {1}{120} \int \left (-\frac {2 \log ((1-x) x)}{\log ^2(x)}+\frac {4 \log ((1-x) x)}{(2-x) \log ^2(x)}-\frac {x \log ((1-x) x)}{\log ^2(x)}\right ) \, dx+\frac {1}{120} \int \left (\frac {\log ((1-x) x)}{\log (x)}+\frac {4 \log ((1-x) x)}{(2-x)^2 \log (x)}+\frac {4 \log ((1-x) x)}{(-2+x) \log (x)}\right ) \, dx+\frac {1}{120} \int \left (-\frac {2 \log ((1-x) x)}{\log (x)}+\frac {4 \log ((1-x) x)}{(2-x) \log (x)}-\frac {x \log ((1-x) x)}{\log (x)}\right ) \, dx-\frac {1}{120} \int \frac {(1-2 x) \text {li}(x)}{(1-x) x} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx-\frac {1}{60} \int \left (\frac {2 \log ((1-x) x)}{(2-x)^2 \log (x)}+\frac {\log ((1-x) x)}{(-2+x) \log (x)}\right ) \, dx+\frac {1}{60} \int \left (\frac {\log ((1-x) x)}{\log (x)}+\frac {2 \log ((1-x) x)}{(-2+x) \log (x)}\right ) \, dx+\frac {1}{60} \int \frac {(1-2 x) \text {li}(x)}{(1-x) x} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx-\frac {1}{40} \int \left (\frac {2 \log ((1-x) x)}{(2-x)^2 \log ^2(x)}+\frac {\log ((1-x) x)}{(-2+x) \log ^2(x)}\right ) \, dx+\frac {1}{40} \int \left (\frac {\log ((1-x) x)}{\log ^2(x)}+\frac {2 \log ((1-x) x)}{(-2+x) \log ^2(x)}\right ) \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx \\ & = -\frac {x}{3}-\frac {1}{120} \log ((1-x) x) \text {li}(x)+2 \left (\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx\right )+\frac {1}{120} \int \frac {\log ((1-x) x)}{\log (x)} \, dx-\frac {1}{120} \int \left (\frac {\text {li}(x)}{-1+x}+\frac {\text {li}(x)}{x}\right ) \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx+\frac {1}{60} \int \left (\frac {\text {li}(x)}{-1+x}+\frac {\text {li}(x)}{x}\right ) \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+2 \left (\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx\right )-\frac {1}{20} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{20} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx \\ & = -\frac {x}{3}+2 \left (\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx\right )-\frac {1}{120} \int \frac {\text {li}(x)}{-1+x} \, dx-\frac {1}{120} \int \frac {\text {li}(x)}{x} \, dx-\frac {1}{120} \int \frac {(1-2 x) \text {li}(x)}{(1-x) x} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\text {li}(x)}{-1+x} \, dx+\frac {1}{60} \int \frac {\text {li}(x)}{x} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+2 \left (\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx\right )-\frac {1}{20} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{20} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx \\ & = -\frac {41 x}{120}+\frac {1}{120} \log (x) \text {li}(x)+2 \left (\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx\right )-\frac {1}{120} \int \frac {\text {li}(x)}{-1+x} \, dx-\frac {1}{120} \int \left (\frac {\text {li}(x)}{-1+x}+\frac {\text {li}(x)}{x}\right ) \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\text {li}(x)}{-1+x} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+2 \left (\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx\right )-\frac {1}{20} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{20} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx \\ & = -\frac {41 x}{120}+\frac {1}{120} \log (x) \text {li}(x)+2 \left (\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx\right )-2 \left (\frac {1}{120} \int \frac {\text {li}(x)}{-1+x} \, dx\right )-\frac {1}{120} \int \frac {\text {li}(x)}{x} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\text {li}(x)}{-1+x} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+2 \left (\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx\right )-\frac {1}{20} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{20} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx \\ & = -\frac {x}{3}+2 \left (\frac {1}{120} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx\right )-2 \left (\frac {1}{120} \int \frac {\text {li}(x)}{-1+x} \, dx\right )-\frac {1}{60} \int \frac {\log ((1-x) x)}{\log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x)^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx-\frac {1}{60} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx+\frac {1}{60} \int \frac {\log ((1-x) x)}{x^2 \log (x)} \, dx+\frac {1}{60} \int \frac {\text {li}(x)}{-1+x} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx-\frac {1}{40} \int \frac {\log ((1-x) x)}{x \log ^2(x)} \, dx+\frac {1}{30} \int \frac {1-2 x}{(-2+x) x^2 \log (x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx+\frac {1}{30} \int \frac {\log ((1-x) x)}{(2-x) \log (x)} \, dx+2 \left (\frac {1}{30} \int \frac {\log ((1-x) x)}{(-2+x) \log (x)} \, dx\right )-\frac {1}{20} \int \frac {\log ((1-x) x)}{(2-x)^2 \log ^2(x)} \, dx+\frac {1}{20} \int \frac {\log ((1-x) x)}{(-2+x) \log ^2(x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=\frac {1}{30} \left (-10 x-\frac {(-1+x) \log (-((-1+x) x))}{(-2+x) x \log (x)}\right ) \]

[In]

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]

[Out]

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

Maple [A] (verified)

Time = 5.25 (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	\(-\frac {\left (-1+x \right ) \ln \left (-1+x \right )}{30 x \left (-2+x \right ) \ln \left (x \right )}-\frac {-i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{3}-i \pi x \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (i x \left (-1+x \right )\right ) \operatorname {csgn}\left (i x \right )+2 i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2}+i \pi x \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{3}-2 i \pi x \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2}-i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-1+x \right )\right )-2 i \pi +i \pi \,\operatorname {csgn}\left (i x \left (-1+x \right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (-1+x \right )\right )+2 i \pi x -i \pi \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+i \pi x \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+i \pi x \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (i x \left (-1+x \right )\right )^{2}+20 x^{3} \ln \left (x \right )-40 x^{2} \ln \left (x \right )+2 x \ln \left (x \right )-2 \ln \left (x \right )}{60 x \left (-2+x \right ) \ln \left (x \right )}\)	\(259\)

[In]

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=_RETURNVERBOSE)

[Out]

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)

none

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=-\frac {{\left (x - 1\right )} \log \left (-x^{2} + x\right ) + 10 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right )}{30 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right )} \]

[In]

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")

[Out]

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

[In]

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)/ln(x)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=-\frac {{\left (10 \, x^{3} - 20 \, x^{2} + x - 1\right )} \log \left (x\right ) + {\left (x - 1\right )} \log \left (-x + 1\right )}{30 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right )} \]

[In]

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")

[Out]

-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)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=-\frac {1}{3} \, x - \frac {{\left (x - 1\right )} \log \left (-x + 1\right )}{30 \, {\left (x^{2} \log \left (x\right ) - 2 \, x \log \left (x\right )\right )}} - \frac {1}{60 \, {\left (x - 2\right )}} - \frac {1}{60 \, x} \]

[In]

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")

[Out]

-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 = 9.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int \frac {\left (-40 x^2+40 x^3-10 x^4\right ) \log ^2(x)+\left (2-3 x+x^2\right ) \log \left (x-x^2\right )+\log (x) \left (-2+5 x-2 x^2+\left (2-2 x+x^2\right ) \log \left (x-x^2\right )\right )}{\left (120 x^2-120 x^3+30 x^4\right ) \log ^2(x)} \, dx=\frac {\frac {\ln \left (x-x^2\right )}{30}-\frac {x\,\ln \left (x-x^2\right )}{30}}{x\,\ln \left (x\right )\,\left (x-2\right )}-\frac {x}{3} \]

[In]

int((log(x)*(5*x - 2*x^2 + log(x - x^2)*(x^2 - 2*x + 2) - 2) - log(x)^2*(40*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)

[Out]

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