\(\int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log (4 x^2+x^3)+(-8-2 x+(-8-2 x) \log (x)+(20 x^2+5 x^3) \log ^2(x)) \log ^2(4 x^2+x^3)}{(4 x+x^2) \log ^2(x) \log (4 x^2+x^3)+((8 x+2 x^2) \log (x)+(8 x^2+22 x^3+5 x^4) \log ^2(x)) \log ^2(4 x^2+x^3)} \, dx\) [8]

Optimal result

Integrand size = 146, antiderivative size = 33 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 x+\frac {2 x+\frac {2+\frac {\log (x)}{\log \left (x^2 (4+x)\right )}}{\log (x)}}{x}\right ) \]

[Out]

ln(5*x+((2+ln(x)/ln(x^2*(4+x)))/ln(x)+2*x)/x)

Rubi [F]

\[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx \]

[In]

Int[((-8 - 3*x)*Log[x]^2 + (-4 - x)*Log[x]^2*Log[4*x^2 + x^3] + (-8 - 2*x + (-8 - 2*x)*Log[x] + (20*x^2 + 5*x^
3)*Log[x]^2)*Log[4*x^2 + x^3]^2)/((4*x + x^2)*Log[x]^2*Log[4*x^2 + x^3] + ((8*x + 2*x^2)*Log[x] + (8*x^2 + 22*
x^3 + 5*x^4)*Log[x]^2)*Log[4*x^2 + x^3]^2),x]

[Out]

Log[2 + 5*x] - Log[Log[x]] + 2*Defer[Int][(2 + 2*x*Log[x] + 5*x^2*Log[x])^(-1), x] - 2*Defer[Int][1/(x*(2 + 2*
x*Log[x] + 5*x^2*Log[x])), x] + 5*Defer[Int][x/(2 + 2*x*Log[x] + 5*x^2*Log[x]), x] - 10*Defer[Int][1/((2 + 5*x
)*(2 + 2*x*Log[x] + 5*x^2*Log[x])), x] - Defer[Int][(8 + 3*x)/(x*(4 + x)*Log[x^2*(4 + x)]), x] + 2*Defer[Int][
1/(x*(2 + 2*x*Log[x] + 5*x^2*Log[x])*(Log[x] + 2*Log[x^2*(4 + x)] + 2*x*Log[x]*Log[x^2*(4 + x)] + 5*x^2*Log[x]
*Log[x^2*(4 + x)])), x] + 2*Defer[Int][Log[x]/(x*(2 + 2*x*Log[x] + 5*x^2*Log[x])*(Log[x] + 2*Log[x^2*(4 + x)]
+ 2*x*Log[x]*Log[x^2*(4 + x)] + 5*x^2*Log[x]*Log[x^2*(4 + x)])), x] - 5*Defer[Int][(x*Log[x]^2)/((2 + 2*x*Log[
x] + 5*x^2*Log[x])*(Log[x] + 2*Log[x^2*(4 + x)] + 2*x*Log[x]*Log[x^2*(4 + x)] + 5*x^2*Log[x]*Log[x^2*(4 + x)])
), x] + 4*Defer[Int][1/(x*(Log[x] + 2*Log[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x] + 2*Defer[I
nt][1/((4 + x)*(Log[x] + 2*Log[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x] - 14*Defer[Int][Log[x]
/(Log[x] + 2*Log[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)]), x] - Defer[Int][Log[x]/(x*(Log[x] + 2*Lo
g[x^2*(4 + x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x] + 15*Defer[Int][(x*Log[x])/(Log[x] + 2*Log[x^2*(4 +
x)] + x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)]), x] + 72*Defer[Int][Log[x]/((4 + x)*(Log[x] + 2*Log[x^2*(4 + x)] +
x*(2 + 5*x)*Log[x]*Log[x^2*(4 + x)])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{x (4+x) \log (x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = \int \left (-\frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}-\frac {(8+3 x) \log (x)}{x (4+x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {\left (-2-2 \log (x)+5 x^2 \log ^2(x)\right ) \log \left (x^2 (4+x)\right )}{x \log (x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx \\ & = -\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {(8+3 x) \log (x)}{x (4+x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \frac {\left (-2-2 \log (x)+5 x^2 \log ^2(x)\right ) \log \left (x^2 (4+x)\right )}{x \log (x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = -\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {(8+3 x) \log (x)}{x (4+x) \log \left (x^2 (4+x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \frac {\left (-2-2 \log (x)+5 x^2 \log ^2(x)\right ) \log \left (x^2 (4+x)\right )}{x \log (x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = -\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \left (\frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )}-\frac {(8+3 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}{x (4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx+\int \left (\frac {-2-2 \log (x)+5 x^2 \log ^2(x)}{x \log (x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}+\frac {2+2 \log (x)-5 x^2 \log ^2(x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx \\ & = \int \frac {-2-2 \log (x)+5 x^2 \log ^2(x)}{x \log (x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx+\int \frac {(8+3 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}{x (4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \frac {2+2 \log (x)-5 x^2 \log ^2(x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = \int \left (\frac {5}{2+5 x}-\frac {1}{x \log (x)}-\frac {4 (1+5 x)}{x (2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}+\frac {2+5 x}{2+2 x \log (x)+5 x^2 \log (x)}\right ) \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \left (\frac {2}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {2 \log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}-\frac {5 x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx+\int \left (\frac {2 \left (2+2 x \log (x)+5 x^2 \log (x)\right )}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {2+2 x \log (x)+5 x^2 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx \\ & = \log (2+5 x)+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {2+2 x \log (x)+5 x^2 \log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-4 \int \frac {1+5 x}{x (2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {2+5 x}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx+\int \frac {2+2 x \log (x)+5 x^2 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = \log (2+5 x)+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \left (\frac {2}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {2 \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )}+\frac {5 x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )}\right ) \, dx-4 \int \left (\frac {1}{2 x \left (2+2 x \log (x)+5 x^2 \log (x)\right )}+\frac {5}{2 (2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )}\right ) \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \left (\frac {2}{2+2 x \log (x)+5 x^2 \log (x)}+\frac {5 x}{2+2 x \log (x)+5 x^2 \log (x)}\right ) \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+\int \left (\frac {2}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {2 x \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}+\frac {5 x^2 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = \log (2+5 x)-\log (\log (x))+2 \int \frac {1}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+2 \int \frac {1}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {x \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {1}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )} \, dx+5 \int \frac {x}{2+2 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x^2 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-10 \int \frac {1}{(2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+10 \int \frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = \log (2+5 x)-\log (\log (x))+2 \int \frac {1}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {1}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {x \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {1}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx+5 \int \frac {x}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+5 \int \frac {x^2 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-10 \int \frac {1}{(2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+10 \int \frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = \log (2+5 x)-\log (\log (x))+2 \int \frac {1}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {1}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \left (\frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )}-\frac {4 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx+4 \int \frac {1}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx+5 \int \frac {x}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+5 \int \left (-\frac {4 \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )}+\frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )}+\frac {16 \log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )}\right ) \, dx-10 \int \frac {1}{(2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+10 \int \frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = \log (2+5 x)-\log (\log (x))+2 \int \frac {1}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+2 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )} \, dx+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {1}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {1}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx+5 \int \frac {x}{2+2 x \log (x)+5 x^2 \log (x)} \, dx+5 \int \frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )} \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-8 \int \frac {\log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-10 \int \frac {1}{(2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+10 \int \frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx-20 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )} \, dx+80 \int \frac {\log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ & = \log (2+5 x)-\log (\log (x))+2 \int \frac {1}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+2 \int \frac {1}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{x \left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {1}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+2 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx+4 \int \frac {1}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+4 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx+5 \int \frac {x}{2+2 x \log (x)+5 x^2 \log (x)} \, dx-5 \int \frac {x \log ^2(x)}{\left (2+2 x \log (x)+5 x^2 \log (x)\right ) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx+5 \int \frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx-8 \int \frac {\log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-10 \int \frac {1}{(2+5 x) \left (2+2 x \log (x)+5 x^2 \log (x)\right )} \, dx+10 \int \frac {x \log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx-20 \int \frac {\log (x)}{\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )} \, dx+80 \int \frac {\log (x)}{(4+x) \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx-\int \frac {8+3 x}{x (4+x) \log \left (x^2 (4+x)\right )} \, dx-\int \frac {\log (x)}{x \left (\log (x)+2 \log \left (x^2 (4+x)\right )+x (2+5 x) \log (x) \log \left (x^2 (4+x)\right )\right )} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(33)=66\).

Time = 55.71 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.27 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log (2+5 x)-\log (x (2+5 x))-\log (\log (x))-\log \left (\log \left (x^2 (4+x)\right )\right )+\log \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right ) \]

[In]

Integrate[((-8 - 3*x)*Log[x]^2 + (-4 - x)*Log[x]^2*Log[4*x^2 + x^3] + (-8 - 2*x + (-8 - 2*x)*Log[x] + (20*x^2
+ 5*x^3)*Log[x]^2)*Log[4*x^2 + x^3]^2)/((4*x + x^2)*Log[x]^2*Log[4*x^2 + x^3] + ((8*x + 2*x^2)*Log[x] + (8*x^2
+ 22*x^3 + 5*x^4)*Log[x]^2)*Log[4*x^2 + x^3]^2),x]

[Out]

Log[2 + 5*x] - Log[x*(2 + 5*x)] - Log[Log[x]] - Log[Log[x^2*(4 + x)]] + Log[Log[x] + 2*Log[x^2*(4 + x)] + 2*x*
Log[x]*Log[x^2*(4 + x)] + 5*x^2*Log[x]*Log[x^2*(4 + x)]]

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.97

method	result	size
parallelrisch	\(-\ln \left (x \right )-\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x^{2} \left (4+x \right )\right )\right )+\ln \left (\ln \left (x \right ) \ln \left (x^{2} \left (4+x \right )\right ) x^{2}+\frac {2 \ln \left (x \right ) \ln \left (x^{2} \left (4+x \right )\right ) x}{5}+\frac {\ln \left (x \right )}{5}+\frac {2 \ln \left (x^{2} \left (4+x \right )\right )}{5}\right )\)	\(65\)
default	\(\text {Expression too large to display}\)	\(675\)
risch	\(\text {Expression too large to display}\)	\(675\)

[In]

int((((5*x^3+20*x^2)*ln(x)^2+(-2*x-8)*ln(x)-2*x-8)*ln(x^3+4*x^2)^2+(-4-x)*ln(x)^2*ln(x^3+4*x^2)+(-3*x-8)*ln(x)
^2)/(((5*x^4+22*x^3+8*x^2)*ln(x)^2+(2*x^2+8*x)*ln(x))*ln(x^3+4*x^2)^2+(x^2+4*x)*ln(x)^2*ln(x^3+4*x^2)),x,metho
d=_RETURNVERBOSE)

[Out]

-ln(x)-ln(ln(x))-ln(ln(x^2*(4+x)))+ln(ln(x)*ln(x^2*(4+x))*x^2+2/5*ln(x)*ln(x^2*(4+x))*x+1/5*ln(x)+2/5*ln(x^2*(
4+x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (33) = 66\).

Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.97 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 \, x + 2\right ) + \log \left (\frac {{\left ({\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2\right )} \log \left (x^{3} + 4 \, x^{2}\right ) + \log \left (x\right )}{{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}\right ) + \log \left (\frac {{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}{5 \, x^{2} + 2 \, x}\right ) - \log \left (\log \left (x^{3} + 4 \, x^{2}\right )\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^2+(-4-x)*log(x)^2*log(x^3+4*x^2)+(-3
*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2)*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^
3+4*x^2)),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate((((5*x**3+20*x**2)*ln(x)**2+(-2*x-8)*ln(x)-2*x-8)*ln(x**3+4*x**2)**2+(-4-x)*ln(x)**2*ln(x**3+4*x**2)
+(-3*x-8)*ln(x)**2)/(((5*x**4+22*x**3+8*x**2)*ln(x)**2+(2*x**2+8*x)*ln(x))*ln(x**3+4*x**2)**2+(x**2+4*x)*ln(x)
**2*ln(x**3+4*x**2)),x)

[Out]

Exception raised: PolynomialError >> 1/(25*_t0**2*x**6 + 120*_t0**2*x**5 + 84*_t0**2*x**4 + 16*_t0**2*x**3 + 2
0*_t0*x**4 + 88*_t0*x**3 + 32*_t0*x**2 + 4*x**2 + 16*x) contains an element of the set of generators.

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (33) = 66\).

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.27 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 \, x + 2\right ) + \log \left (\frac {2 \, {\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right )^{2} + {\left ({\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2\right )} \log \left (x + 4\right ) + 5 \, \log \left (x\right )}{{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}\right ) + \log \left (\frac {{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}{5 \, x^{2} + 2 \, x}\right ) - \log \left (\log \left (x + 4\right ) + 2 \, \log \left (x\right )\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^2+(-4-x)*log(x)^2*log(x^3+4*x^2)+(-3
*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2)*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^
3+4*x^2)),x, algorithm="maxima")

[Out]

log(5*x + 2) + log((2*(5*x^2 + 2*x)*log(x)^2 + ((5*x^2 + 2*x)*log(x) + 2)*log(x + 4) + 5*log(x))/((5*x^2 + 2*x
)*log(x) + 2)) + log(((5*x^2 + 2*x)*log(x) + 2)/(5*x^2 + 2*x)) - log(log(x + 4) + 2*log(x)) - log(log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (33) = 66\).

Time = 0.53 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 \, x^{2} \log \left (x + 4\right ) \log \left (x\right ) + 10 \, x^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x + 4\right ) \log \left (x\right ) + 4 \, x \log \left (x\right )^{2} + 2 \, \log \left (x + 4\right ) + 5 \, \log \left (x\right )\right ) - \log \left (x\right ) - \log \left (\log \left (x + 4\right ) + 2 \, \log \left (x\right )\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^2+(-4-x)*log(x)^2*log(x^3+4*x^2)+(-3
*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2)*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^
3+4*x^2)),x, algorithm="giac")

[Out]

log(5*x^2*log(x + 4)*log(x) + 10*x^2*log(x)^2 + 2*x*log(x + 4)*log(x) + 4*x*log(x)^2 + 2*log(x + 4) + 5*log(x)
) - log(x) - log(log(x + 4) + 2*log(x)) - log(log(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\int -\frac {{\ln \left (x^3+4\,x^2\right )}^2\,\left (\left (-5\,x^3-20\,x^2\right )\,{\ln \left (x\right )}^2+\left (2\,x+8\right )\,\ln \left (x\right )+2\,x+8\right )+{\ln \left (x\right )}^2\,\left (3\,x+8\right )+\ln \left (x^3+4\,x^2\right )\,{\ln \left (x\right )}^2\,\left (x+4\right )}{{\ln \left (x^3+4\,x^2\right )}^2\,\left (\left (5\,x^4+22\,x^3+8\,x^2\right )\,{\ln \left (x\right )}^2+\left (2\,x^2+8\,x\right )\,\ln \left (x\right )\right )+\ln \left (x^3+4\,x^2\right )\,{\ln \left (x\right )}^2\,\left (x^2+4\,x\right )} \,d x \]

[In]

int(-(log(4*x^2 + x^3)^2*(2*x - log(x)^2*(20*x^2 + 5*x^3) + log(x)*(2*x + 8) + 8) + log(x)^2*(3*x + 8) + log(4
*x^2 + x^3)*log(x)^2*(x + 4))/(log(4*x^2 + x^3)^2*(log(x)^2*(8*x^2 + 22*x^3 + 5*x^4) + log(x)*(8*x + 2*x^2)) +
log(4*x^2 + x^3)*log(x)^2*(4*x + x^2)),x)

[Out]

int(-(log(4*x^2 + x^3)^2*(2*x - log(x)^2*(20*x^2 + 5*x^3) + log(x)*(2*x + 8) + 8) + log(x)^2*(3*x + 8) + log(4
*x^2 + x^3)*log(x)^2*(x + 4))/(log(4*x^2 + x^3)^2*(log(x)^2*(8*x^2 + 22*x^3 + 5*x^4) + log(x)*(8*x + 2*x^2)) +
log(4*x^2 + x^3)*log(x)^2*(4*x + x^2)), x)