\(\int \frac {-x-x^2+x^3-2 x^4+(x^2-2 x^3) \log (x)+(-x+2 x^3+(-1+2 x^2) \log (x)) \log (x^2+x \log (x))+(1+2 x-x^2+3 x^3+(1-x+3 x^2) \log (x)+(-3 x^2-3 x \log (x)) \log (x^2+x \log (x))) \log (x-\log (x^2+x \log (x)))+(-x^2-x \log (x)+(x+\log (x)) \log (x^2+x \log (x))) \log ^2(x-\log (x^2+x \log (x)))}{x^3-x^5+(x^2-x^4) \log (x)+(-x^2+x^4+(-x+x^3) \log (x)) \log (x^2+x \log (x))+(2 x^4+2 x^3 \log (x)+(-2 x^3-2 x^2 \log (x)) \log (x^2+x \log (x))) \log (x-\log (x^2+x \log (x)))+(-x^3-x^2 \log (x)+(x^2+x \log (x)) \log (x^2+x \log (x))) \log ^2(x-\log (x^2+x \log (x)))} \, dx\) [7578]

Optimal result

Integrand size = 316, antiderivative size = 31 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {1}{2} \log \left (x \left (-x+x (x-\log (x-\log (x (x+\log (x)))))^2\right )\right ) \]

[Out]

1/2*ln((x*(x-ln(x-ln((x+ln(x))*x)))^2-x)*x)

Rubi [F]

\[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx \]

[In]

Int[(-x - x^2 + x^3 - 2*x^4 + (x^2 - 2*x^3)*Log[x] + (-x + 2*x^3 + (-1 + 2*x^2)*Log[x])*Log[x^2 + x*Log[x]] +
(1 + 2*x - x^2 + 3*x^3 + (1 - x + 3*x^2)*Log[x] + (-3*x^2 - 3*x*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 +
x*Log[x]]] + (-x^2 - x*Log[x] + (x + Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]]^2)/(x^3 - x^5
+ (x^2 - x^4)*Log[x] + (-x^2 + x^4 + (-x + x^3)*Log[x])*Log[x^2 + x*Log[x]] + (2*x^4 + 2*x^3*Log[x] + (-2*x^3
- 2*x^2*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]] + (-x^3 - x^2*Log[x] + (x^2 + x*Log[x])*Log[
x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]]^2),x]

[Out]

Log[x] - Defer[Int][(-1 + x - Log[x - Log[x*(x + Log[x])]])^(-1), x] + Defer[Int][1/(x*(-1 + x - Log[x - Log[x
*(x + Log[x])]])), x]/2 + Defer[Int][x/(-1 + x - Log[x - Log[x*(x + Log[x])]]), x]/2 + Defer[Int][1/((x + Log[
x])*(x - Log[x*(x + Log[x])])*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x] + Defer[Int][1/(x*(x + Log[x])*(x -
Log[x*(x + Log[x])])*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 - Defer[Int][x/((x + Log[x])*(x - Log[x*(
x + Log[x])])*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x] + (3*Defer[Int][x^2/((x + Log[x])*(x - Log[x*(x + L
og[x])])*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x])/2 - Defer[Int][x^3/((x + Log[x])*(x - Log[x*(x + Log[x]
)])*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 - Defer[Int][Log[x]/((x + Log[x])*(x - Log[x*(x + Log[x])])
*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x] + Defer[Int][Log[x]/(x*(x + Log[x])*(x - Log[x*(x + Log[x])])*(-
1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 + (3*Defer[Int][(x*Log[x])/((x + Log[x])*(x - Log[x*(x + Log[x])]
)*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x])/2 - Defer[Int][(x^2*Log[x])/((x + Log[x])*(x - Log[x*(x + Log[
x])])*(-1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 - Defer[Int][(1 + x - Log[x - Log[x*(x + Log[x])]])^(-1),
x] - Defer[Int][1/(x*(1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 - Defer[Int][x/(1 + x - Log[x - Log[x*(x +
Log[x])]]), x]/2 + Defer[Int][1/((x + Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]]
)), x] + Defer[Int][1/(x*(x + Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2
+ (3*Defer[Int][x^2/((x + Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x])/2 + D
efer[Int][x^3/((x + Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 + Defer[In
t][Log[x]/(x*(x + Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 + (3*Defer[I
nt][(x*Log[x])/((x + Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x])/2 + Defer[
Int][(x^2*Log[x])/((x + Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 + Defe
r[Int][(x*Log[x^2 + x*Log[x]])/((x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x] + (3*Def
er[Int][(x*Log[x^2 + x*Log[x]])/((-x - Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]]
)), x])/2 + (3*Defer[Int][(x^2*Log[x^2 + x*Log[x]])/((-x - Log[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x -
Log[x*(x + Log[x])]])), x])/2 + (3*Defer[Int][(Log[x]*Log[x^2 + x*Log[x]])/((-x - Log[x])*(x - Log[x*(x + Log[
x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x])/2 + (3*Defer[Int][(x*Log[x]*Log[x^2 + x*Log[x]])/((-x - Log
[x])*(x - Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x])/2 + Defer[Int][Log[x^2 + x*Log[x]]
/(x*(-x + Log[x*(x + Log[x])])*(1 + x - Log[x - Log[x*(x + Log[x])]])), x]/2 + Defer[Int][(x*Log[x^2 + x*Log[x
]])/((x - Log[x*(x + Log[x])])*(1 - x + Log[x - Log[x*(x + Log[x])]])), x] - (3*Defer[Int][(x*Log[x^2 + x*Log[
x]])/((-x - Log[x])*(x - Log[x*(x + Log[x])])*(1 - x + Log[x - Log[x*(x + Log[x])]])), x])/2 - (3*Defer[Int][(
Log[x]*Log[x^2 + x*Log[x]])/((-x - Log[x])*(x - Log[x*(x + Log[x])])*(1 - x + Log[x - Log[x*(x + Log[x])]])),
x])/2 - (3*Defer[Int][(x^2*Log[x^2 + x*Log[x]])/((x + Log[x])*(x - Log[x*(x + Log[x])])*(1 - x + Log[x - Log[x
*(x + Log[x])]])), x])/2 - (3*Defer[Int][(x*Log[x]*Log[x^2 + x*Log[x]])/((x + Log[x])*(x - Log[x*(x + Log[x])]
)*(1 - x + Log[x - Log[x*(x + Log[x])]])), x])/2 + Defer[Int][Log[x^2 + x*Log[x]]/(x*(-x + Log[x*(x + Log[x])]
)*(1 - x + Log[x - Log[x*(x + Log[x])]])), x]/2

Rubi steps \begin{align*} \text {integral}& = \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x (x+\log (x)) (x-\log (x (x+\log (x)))) \left (1-x^2+2 x \log (x-\log (x (x+\log (x))))-\log ^2(x-\log (x (x+\log (x))))\right )} \, dx \\ & = \int \left (\frac {1}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {x}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x^2}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {2 x^3}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {x (-1+2 x) \log (x)}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}-\frac {\left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right ) \log (x-\log (x (x+\log (x))))}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {\log ^2(x-\log (x (x+\log (x))))}{x (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{x (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x))))) (1-x+\log (x-\log (x (x+\log (x)))))}\right ) \, dx \\ & = 2 \int \frac {x^3}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {1}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {x}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\int \frac {x^2}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {x (-1+2 x) \log (x)}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\int \frac {\left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right ) \log (x-\log (x (x+\log (x))))}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {\log ^2(x-\log (x (x+\log (x))))}{x (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{x (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x))))) (1-x+\log (x-\log (x (x+\log (x)))))} \, dx \\ & = 2 \int \left (\frac {x^3}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x^3}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {1}{x}+\frac {(-1+x)^2}{2 x (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {(1+x)^2}{2 x (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {1}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {1}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {x}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx-\int \left (\frac {x^2}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x^2}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {x (-1+2 x) \log (x)}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x (-1+2 x) \log (x)}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx-\int \left (\frac {(-1+x) \left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right )}{2 x (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {(1+x) \left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right )}{2 x (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{2 x (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{2 x (x-\log (x (x+\log (x)))) (1-x+\log (x-\log (x (x+\log (x)))))}\right ) \, dx \\ & = \log (x)+\frac {1}{2} \int \frac {(-1+x)^2}{x (-1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\frac {1}{2} \int \frac {1}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\frac {1}{2} \int \frac {x}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\frac {1}{2} \int \frac {x^2}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\frac {1}{2} \int \frac {x (-1+2 x) \log (x)}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\frac {1}{2} \int \frac {(-1+x) \left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right )}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\frac {1}{2} \int \frac {(1+x)^2}{x (1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\frac {1}{2} \int \frac {1}{(x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\frac {1}{2} \int \frac {x}{(x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\frac {1}{2} \int \frac {x^2}{(x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\frac {1}{2} \int \frac {x (-1+2 x) \log (x)}{(x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\frac {1}{2} \int \frac {(1+x) \left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right )}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\frac {1}{2} \int \frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{x (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\frac {1}{2} \int \frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{x (x-\log (x (x+\log (x)))) (1-x+\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {x^3}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\int \frac {x^3}{(x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\log (x)+\frac {1}{2} \log \left (1-x^2+2 x \log (x-\log (x (x+\log (x))))-\log ^2(x-\log (x (x+\log (x))))\right ) \]

[In]

Integrate[(-x - x^2 + x^3 - 2*x^4 + (x^2 - 2*x^3)*Log[x] + (-x + 2*x^3 + (-1 + 2*x^2)*Log[x])*Log[x^2 + x*Log[
x]] + (1 + 2*x - x^2 + 3*x^3 + (1 - x + 3*x^2)*Log[x] + (-3*x^2 - 3*x*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log
[x^2 + x*Log[x]]] + (-x^2 - x*Log[x] + (x + Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]]^2)/(x^3
- x^5 + (x^2 - x^4)*Log[x] + (-x^2 + x^4 + (-x + x^3)*Log[x])*Log[x^2 + x*Log[x]] + (2*x^4 + 2*x^3*Log[x] + (-
2*x^3 - 2*x^2*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]] + (-x^3 - x^2*Log[x] + (x^2 + x*Log[x]
)*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]]^2),x]

[Out]

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

Maple [A] (verified)

Time = 130.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42

method	result	size
parallelrisch	\(\frac {\ln \left (x -\ln \left (x -\ln \left (\left (x +\ln \left (x \right )\right ) x \right )\right )-1\right )}{2}+\frac {\ln \left (x -\ln \left (x -\ln \left (\left (x +\ln \left (x \right )\right ) x \right )\right )+1\right )}{2}+\ln \left (x \right )\)	\(44\)
default	\(\ln \left (x \right )+\frac {\ln \left (x^{2}-2 x \ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )+\ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )^{2}-1\right )}{2}\)	\(145\)
risch	\(\ln \left (x \right )+\frac {\ln \left (x^{2}-2 x \ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )+\ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )^{2}-1\right )}{2}\)	\(145\)

[In]

int((((x+ln(x))*ln(x*ln(x)+x^2)-x*ln(x)-x^2)*ln(-ln(x*ln(x)+x^2)+x)^2+((-3*x*ln(x)-3*x^2)*ln(x*ln(x)+x^2)+(3*x
^2-x+1)*ln(x)+3*x^3-x^2+2*x+1)*ln(-ln(x*ln(x)+x^2)+x)+((2*x^2-1)*ln(x)+2*x^3-x)*ln(x*ln(x)+x^2)+(-2*x^3+x^2)*l
n(x)-2*x^4+x^3-x^2-x)/(((x*ln(x)+x^2)*ln(x*ln(x)+x^2)-x^2*ln(x)-x^3)*ln(-ln(x*ln(x)+x^2)+x)^2+((-2*x^2*ln(x)-2
*x^3)*ln(x*ln(x)+x^2)+2*x^3*ln(x)+2*x^4)*ln(-ln(x*ln(x)+x^2)+x)+((x^3-x)*ln(x)+x^4-x^2)*ln(x*ln(x)+x^2)+(-x^4+
x^2)*ln(x)-x^5+x^3),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(x-ln(x-ln((x+ln(x))*x))-1)+1/2*ln(x-ln(x-ln((x+ln(x))*x))+1)+ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {1}{2} \, \log \left (x^{2} - 2 \, x \log \left (x - \log \left (x^{2} + x \log \left (x\right )\right )\right ) + \log \left (x - \log \left (x^{2} + x \log \left (x\right )\right )\right )^{2} - 1\right ) + \log \left (x\right ) \]

[In]

integrate((((x+log(x))*log(x*log(x)+x^2)-x*log(x)-x^2)*log(-log(x*log(x)+x^2)+x)^2+((-3*x*log(x)-3*x^2)*log(x*
log(x)+x^2)+(3*x^2-x+1)*log(x)+3*x^3-x^2+2*x+1)*log(-log(x*log(x)+x^2)+x)+((2*x^2-1)*log(x)+2*x^3-x)*log(x*log
(x)+x^2)+(-2*x^3+x^2)*log(x)-2*x^4+x^3-x^2-x)/(((x*log(x)+x^2)*log(x*log(x)+x^2)-x^2*log(x)-x^3)*log(-log(x*lo
g(x)+x^2)+x)^2+((-2*x^2*log(x)-2*x^3)*log(x*log(x)+x^2)+2*x^3*log(x)+2*x^4)*log(-log(x*log(x)+x^2)+x)+((x^3-x)
*log(x)+x^4-x^2)*log(x*log(x)+x^2)+(-x^4+x^2)*log(x)-x^5+x^3),x, algorithm="fricas")

[Out]

1/2*log(x^2 - 2*x*log(x - log(x^2 + x*log(x))) + log(x - log(x^2 + x*log(x)))^2 - 1) + log(x)

Sympy [A] (verification not implemented)

Time = 4.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\log {\left (x \right )} + \frac {\log {\left (x^{2} - 2 x \log {\left (x - \log {\left (x^{2} + x \log {\left (x \right )} \right )} \right )} + \log {\left (x - \log {\left (x^{2} + x \log {\left (x \right )} \right )} \right )}^{2} - 1 \right )}}{2} \]

[In]

integrate((((x+ln(x))*ln(x*ln(x)+x**2)-x*ln(x)-x**2)*ln(-ln(x*ln(x)+x**2)+x)**2+((-3*x*ln(x)-3*x**2)*ln(x*ln(x
)+x**2)+(3*x**2-x+1)*ln(x)+3*x**3-x**2+2*x+1)*ln(-ln(x*ln(x)+x**2)+x)+((2*x**2-1)*ln(x)+2*x**3-x)*ln(x*ln(x)+x
**2)+(-2*x**3+x**2)*ln(x)-2*x**4+x**3-x**2-x)/(((x*ln(x)+x**2)*ln(x*ln(x)+x**2)-x**2*ln(x)-x**3)*ln(-ln(x*ln(x
)+x**2)+x)**2+((-2*x**2*ln(x)-2*x**3)*ln(x*ln(x)+x**2)+2*x**3*ln(x)+2*x**4)*ln(-ln(x*ln(x)+x**2)+x)+((x**3-x)*
ln(x)+x**4-x**2)*ln(x*ln(x)+x**2)+(-x**4+x**2)*ln(x)-x**5+x**3),x)

[Out]

log(x) + log(x**2 - 2*x*log(x - log(x**2 + x*log(x))) + log(x - log(x**2 + x*log(x)))**2 - 1)/2

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\log \left (x\right ) + \frac {1}{2} \, \log \left (-x + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) + 1\right ) + \frac {1}{2} \, \log \left (-x + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) - 1\right ) \]

[In]

integrate((((x+log(x))*log(x*log(x)+x^2)-x*log(x)-x^2)*log(-log(x*log(x)+x^2)+x)^2+((-3*x*log(x)-3*x^2)*log(x*
log(x)+x^2)+(3*x^2-x+1)*log(x)+3*x^3-x^2+2*x+1)*log(-log(x*log(x)+x^2)+x)+((2*x^2-1)*log(x)+2*x^3-x)*log(x*log
(x)+x^2)+(-2*x^3+x^2)*log(x)-2*x^4+x^3-x^2-x)/(((x*log(x)+x^2)*log(x*log(x)+x^2)-x^2*log(x)-x^3)*log(-log(x*lo
g(x)+x^2)+x)^2+((-2*x^2*log(x)-2*x^3)*log(x*log(x)+x^2)+2*x^3*log(x)+2*x^4)*log(-log(x*log(x)+x^2)+x)+((x^3-x)
*log(x)+x^4-x^2)*log(x*log(x)+x^2)+(-x^4+x^2)*log(x)-x^5+x^3),x, algorithm="maxima")

[Out]

log(x) + 1/2*log(-x + log(x - log(x + log(x)) - log(x)) + 1) + 1/2*log(-x + log(x - log(x + log(x)) - log(x))
- 1)

Giac [A] (verification not implemented)

none

Time = 1.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {1}{2} \, \log \left (x^{2} - 2 \, x \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right )^{2} - 1\right ) + \log \left (x\right ) \]

[In]

integrate((((x+log(x))*log(x*log(x)+x^2)-x*log(x)-x^2)*log(-log(x*log(x)+x^2)+x)^2+((-3*x*log(x)-3*x^2)*log(x*
log(x)+x^2)+(3*x^2-x+1)*log(x)+3*x^3-x^2+2*x+1)*log(-log(x*log(x)+x^2)+x)+((2*x^2-1)*log(x)+2*x^3-x)*log(x*log
(x)+x^2)+(-2*x^3+x^2)*log(x)-2*x^4+x^3-x^2-x)/(((x*log(x)+x^2)*log(x*log(x)+x^2)-x^2*log(x)-x^3)*log(-log(x*lo
g(x)+x^2)+x)^2+((-2*x^2*log(x)-2*x^3)*log(x*log(x)+x^2)+2*x^3*log(x)+2*x^4)*log(-log(x*log(x)+x^2)+x)+((x^3-x)
*log(x)+x^4-x^2)*log(x*log(x)+x^2)+(-x^4+x^2)*log(x)-x^5+x^3),x, algorithm="giac")

[Out]

1/2*log(x^2 - 2*x*log(x - log(x + log(x)) - log(x)) + log(x - log(x + log(x)) - log(x))^2 - 1) + log(x)

Mupad [B] (verification not implemented)

Time = 14.73 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {\ln \left (x^2-2\,x\,\ln \left (x-\ln \left (x\,\left (x+\ln \left (x\right )\right )\right )\right )+{\ln \left (x-\ln \left (x\,\left (x+\ln \left (x\right )\right )\right )\right )}^2-1\right )}{2}+\ln \left (x\right ) \]

[In]

int((x - log(x*log(x) + x^2)*(2*x^3 - x + log(x)*(2*x^2 - 1)) + log(x - log(x*log(x) + x^2))^2*(x*log(x) - log
(x*log(x) + x^2)*(x + log(x)) + x^2) - log(x)*(x^2 - 2*x^3) - log(x - log(x*log(x) + x^2))*(2*x - log(x*log(x)
+ x^2)*(3*x*log(x) + 3*x^2) + log(x)*(3*x^2 - x + 1) - x^2 + 3*x^3 + 1) + x^2 - x^3 + 2*x^4)/(log(x*log(x) +
x^2)*(log(x)*(x - x^3) + x^2 - x^4) - log(x - log(x*log(x) + x^2))*(2*x^3*log(x) - log(x*log(x) + x^2)*(2*x^2*
log(x) + 2*x^3) + 2*x^4) - log(x)*(x^2 - x^4) + log(x - log(x*log(x) + x^2))^2*(x^2*log(x) - log(x*log(x) + x^
2)*(x*log(x) + x^2) + x^3) - x^3 + x^5),x)

[Out]

log(log(x - log(x*(x + log(x))))^2 - 2*x*log(x - log(x*(x + log(x)))) + x^2 - 1)/2 + log(x)