\(\int \frac {(10-10 x+x^2) \log (x)+(20 x-4 x^2+(40 x-8 x^2) \log (x)) \log (x^2 \log (x))+(20 x-2 x^2) \log (x) \log ^2(x^2 \log (x))+(-20 x+4 x^2+(-40 x+8 x^2) \log (x)) \log ^3(x^2 \log (x))+(-10 x+x^2) \log (x) \log ^4(x^2 \log (x))}{(4 x^2-4 x^3+x^4) \log (x)+(8 x^3-4 x^4) \log (x) \log ^2(x^2 \log (x))+(-4 x^3+6 x^4) \log (x) \log ^4(x^2 \log (x))-4 x^4 \log (x) \log ^6(x^2 \log (x))+x^4 \log (x) \log ^8(x^2 \log (x))} \, dx\) [6581]

Optimal result

Integrand size = 210, antiderivative size = 28 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\frac {-5+x}{2 x-\left (x-x \log ^2\left (x^2 \log (x)\right )\right )^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx \]

[In]

Int[((10 - 10*x + x^2)*Log[x] + (20*x - 4*x^2 + (40*x - 8*x^2)*Log[x])*Log[x^2*Log[x]] + (20*x - 2*x^2)*Log[x]
*Log[x^2*Log[x]]^2 + (-20*x + 4*x^2 + (-40*x + 8*x^2)*Log[x])*Log[x^2*Log[x]]^3 + (-10*x + x^2)*Log[x]*Log[x^2
*Log[x]]^4)/((4*x^2 - 4*x^3 + x^4)*Log[x] + (8*x^3 - 4*x^4)*Log[x]*Log[x^2*Log[x]]^2 + (-4*x^3 + 6*x^4)*Log[x]
*Log[x^2*Log[x]]^4 - 4*x^4*Log[x]*Log[x^2*Log[x]]^6 + x^4*Log[x]*Log[x^2*Log[x]]^8),x]

[Out]

-10*Defer[Int][1/(x^2*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4)^2), x] + 2*Defer[Int][1/(x*(-2 +
x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4)^2), x] - 8*Defer[Int][Log[x^2*Log[x]]/(-2 + x - 2*x*Log[x^2*L
og[x]]^2 + x*Log[x^2*Log[x]]^4)^2, x] + 40*Defer[Int][Log[x^2*Log[x]]/(x*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*L
og[x^2*Log[x]]^4)^2), x] - 4*Defer[Int][Log[x^2*Log[x]]/(Log[x]*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Lo
g[x]]^4)^2), x] + 20*Defer[Int][Log[x^2*Log[x]]/(x*Log[x]*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^
4)^2), x] + 8*Defer[Int][Log[x^2*Log[x]]^3/(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4)^2, x] - 40*D
efer[Int][Log[x^2*Log[x]]^3/(x*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4)^2), x] + 4*Defer[Int][Lo
g[x^2*Log[x]]^3/(Log[x]*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4)^2), x] - 20*Defer[Int][Log[x^2*
Log[x]]^3/(x*Log[x]*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4)^2), x] - 10*Defer[Int][1/(x^2*(-2 +
x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4)), x] + Defer[Int][1/(x*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*L
og[x^2*Log[x]]^4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4 (-5+x) x \log \left (x^2 \log (x)\right ) \left (-1+\log ^2\left (x^2 \log (x)\right )\right )+\log (x) \left (10-10 x+x^2-8 (-5+x) x \log \left (x^2 \log (x)\right )-2 (-10+x) x \log ^2\left (x^2 \log (x)\right )+8 (-5+x) x \log ^3\left (x^2 \log (x)\right )+(-10+x) x \log ^4\left (x^2 \log (x)\right )\right )}{x^2 \log (x) \left (2-x+2 x \log ^2\left (x^2 \log (x)\right )-x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx \\ & = \int \left (\frac {2 (-5+x) \left (\log (x)-2 x \log \left (x^2 \log (x)\right )-4 x \log (x) \log \left (x^2 \log (x)\right )+2 x \log ^3\left (x^2 \log (x)\right )+4 x \log (x) \log ^3\left (x^2 \log (x)\right )\right )}{x^2 \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}+\frac {-10+x}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )}\right ) \, dx \\ & = 2 \int \frac {(-5+x) \left (\log (x)-2 x \log \left (x^2 \log (x)\right )-4 x \log (x) \log \left (x^2 \log (x)\right )+2 x \log ^3\left (x^2 \log (x)\right )+4 x \log (x) \log ^3\left (x^2 \log (x)\right )\right )}{x^2 \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx+\int \frac {-10+x}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \, dx \\ & = 2 \int \left (-\frac {5 \left (\log (x)-2 x \log \left (x^2 \log (x)\right )-4 x \log (x) \log \left (x^2 \log (x)\right )+2 x \log ^3\left (x^2 \log (x)\right )+4 x \log (x) \log ^3\left (x^2 \log (x)\right )\right )}{x^2 \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}+\frac {\log (x)-2 x \log \left (x^2 \log (x)\right )-4 x \log (x) \log \left (x^2 \log (x)\right )+2 x \log ^3\left (x^2 \log (x)\right )+4 x \log (x) \log ^3\left (x^2 \log (x)\right )}{x \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}\right ) \, dx+\int \left (-\frac {10}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )}+\frac {1}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )}\right ) \, dx \\ & = 2 \int \frac {\log (x)-2 x \log \left (x^2 \log (x)\right )-4 x \log (x) \log \left (x^2 \log (x)\right )+2 x \log ^3\left (x^2 \log (x)\right )+4 x \log (x) \log ^3\left (x^2 \log (x)\right )}{x \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-10 \int \frac {\log (x)-2 x \log \left (x^2 \log (x)\right )-4 x \log (x) \log \left (x^2 \log (x)\right )+2 x \log ^3\left (x^2 \log (x)\right )+4 x \log (x) \log ^3\left (x^2 \log (x)\right )}{x^2 \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-10 \int \frac {1}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \, dx+\int \frac {1}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \, dx \\ & = 2 \int \left (\frac {1}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}-\frac {4 \log \left (x^2 \log (x)\right )}{\left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}-\frac {2 \log \left (x^2 \log (x)\right )}{\log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}+\frac {4 \log ^3\left (x^2 \log (x)\right )}{\left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}+\frac {2 \log ^3\left (x^2 \log (x)\right )}{\log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}\right ) \, dx-10 \int \frac {1}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \, dx-10 \int \left (\frac {1}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}-\frac {4 \log \left (x^2 \log (x)\right )}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}-\frac {2 \log \left (x^2 \log (x)\right )}{x \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}+\frac {4 \log ^3\left (x^2 \log (x)\right )}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}+\frac {2 \log ^3\left (x^2 \log (x)\right )}{x \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2}\right ) \, dx+\int \frac {1}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \, dx \\ & = 2 \int \frac {1}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-4 \int \frac {\log \left (x^2 \log (x)\right )}{\log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx+4 \int \frac {\log ^3\left (x^2 \log (x)\right )}{\log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-8 \int \frac {\log \left (x^2 \log (x)\right )}{\left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx+8 \int \frac {\log ^3\left (x^2 \log (x)\right )}{\left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-10 \int \frac {1}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-10 \int \frac {1}{x^2 \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \, dx+20 \int \frac {\log \left (x^2 \log (x)\right )}{x \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-20 \int \frac {\log ^3\left (x^2 \log (x)\right )}{x \log (x) \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx+40 \int \frac {\log \left (x^2 \log (x)\right )}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx-40 \int \frac {\log ^3\left (x^2 \log (x)\right )}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )^2} \, dx+\int \frac {1}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\frac {5-x}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \]

[In]

Integrate[((10 - 10*x + x^2)*Log[x] + (20*x - 4*x^2 + (40*x - 8*x^2)*Log[x])*Log[x^2*Log[x]] + (20*x - 2*x^2)*
Log[x]*Log[x^2*Log[x]]^2 + (-20*x + 4*x^2 + (-40*x + 8*x^2)*Log[x])*Log[x^2*Log[x]]^3 + (-10*x + x^2)*Log[x]*L
og[x^2*Log[x]]^4)/((4*x^2 - 4*x^3 + x^4)*Log[x] + (8*x^3 - 4*x^4)*Log[x]*Log[x^2*Log[x]]^2 + (-4*x^3 + 6*x^4)*
Log[x]*Log[x^2*Log[x]]^4 - 4*x^4*Log[x]*Log[x^2*Log[x]]^6 + x^4*Log[x]*Log[x^2*Log[x]]^8),x]

[Out]

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

Maple [F(-1)]

Timed out.

\[\int \frac {\left (x^{2}-10 x \right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{4}+\left (\left (8 x^{2}-40 x \right ) \ln \left (x \right )+4 x^{2}-20 x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{3}+\left (-2 x^{2}+20 x \right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{2}+\left (\left (-8 x^{2}+40 x \right ) \ln \left (x \right )-4 x^{2}+20 x \right ) \ln \left (x^{2} \ln \left (x \right )\right )+\left (x^{2}-10 x +10\right ) \ln \left (x \right )}{x^{4} \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{8}-4 x^{4} \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{6}+\left (6 x^{4}-4 x^{3}\right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{4}+\left (-4 x^{4}+8 x^{3}\right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{2}+\left (x^{4}-4 x^{3}+4 x^{2}\right ) \ln \left (x \right )}d x\]

[In]

int(((x^2-10*x)*ln(x)*ln(x^2*ln(x))^4+((8*x^2-40*x)*ln(x)+4*x^2-20*x)*ln(x^2*ln(x))^3+(-2*x^2+20*x)*ln(x)*ln(x
^2*ln(x))^2+((-8*x^2+40*x)*ln(x)-4*x^2+20*x)*ln(x^2*ln(x))+(x^2-10*x+10)*ln(x))/(x^4*ln(x)*ln(x^2*ln(x))^8-4*x
^4*ln(x)*ln(x^2*ln(x))^6+(6*x^4-4*x^3)*ln(x)*ln(x^2*ln(x))^4+(-4*x^4+8*x^3)*ln(x)*ln(x^2*ln(x))^2+(x^4-4*x^3+4
*x^2)*ln(x)),x)

[Out]

int(((x^2-10*x)*ln(x)*ln(x^2*ln(x))^4+((8*x^2-40*x)*ln(x)+4*x^2-20*x)*ln(x^2*ln(x))^3+(-2*x^2+20*x)*ln(x)*ln(x
^2*ln(x))^2+((-8*x^2+40*x)*ln(x)-4*x^2+20*x)*ln(x^2*ln(x))+(x^2-10*x+10)*ln(x))/(x^4*ln(x)*ln(x^2*ln(x))^8-4*x
^4*ln(x)*ln(x^2*ln(x))^6+(6*x^4-4*x^3)*ln(x)*ln(x^2*ln(x))^4+(-4*x^4+8*x^3)*ln(x)*ln(x^2*ln(x))^2+(x^4-4*x^3+4
*x^2)*ln(x)),x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=-\frac {x - 5}{x^{2} \log \left (x^{2} \log \left (x\right )\right )^{4} - 2 \, x^{2} \log \left (x^{2} \log \left (x\right )\right )^{2} + x^{2} - 2 \, x} \]

[In]

integrate(((x^2-10*x)*log(x)*log(x^2*log(x))^4+((8*x^2-40*x)*log(x)+4*x^2-20*x)*log(x^2*log(x))^3+(-2*x^2+20*x
)*log(x)*log(x^2*log(x))^2+((-8*x^2+40*x)*log(x)-4*x^2+20*x)*log(x^2*log(x))+(x^2-10*x+10)*log(x))/(x^4*log(x)
*log(x^2*log(x))^8-4*x^4*log(x)*log(x^2*log(x))^6+(6*x^4-4*x^3)*log(x)*log(x^2*log(x))^4+(-4*x^4+8*x^3)*log(x)
*log(x^2*log(x))^2+(x^4-4*x^3+4*x^2)*log(x)),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\frac {5 - x}{x^{2} \log {\left (x^{2} \log {\left (x \right )} \right )}^{4} - 2 x^{2} \log {\left (x^{2} \log {\left (x \right )} \right )}^{2} + x^{2} - 2 x} \]

[In]

integrate(((x**2-10*x)*ln(x)*ln(x**2*ln(x))**4+((8*x**2-40*x)*ln(x)+4*x**2-20*x)*ln(x**2*ln(x))**3+(-2*x**2+20
*x)*ln(x)*ln(x**2*ln(x))**2+((-8*x**2+40*x)*ln(x)-4*x**2+20*x)*ln(x**2*ln(x))+(x**2-10*x+10)*ln(x))/(x**4*ln(x
)*ln(x**2*ln(x))**8-4*x**4*ln(x)*ln(x**2*ln(x))**6+(6*x**4-4*x**3)*ln(x)*ln(x**2*ln(x))**4+(-4*x**4+8*x**3)*ln
(x)*ln(x**2*ln(x))**2+(x**4-4*x**3+4*x**2)*ln(x)),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (28) = 56\).

Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=-\frac {x - 5}{16 \, x^{2} \log \left (x\right )^{4} + 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + x^{2} \log \left (\log \left (x\right )\right )^{4} - 8 \, x^{2} \log \left (x\right )^{2} + 2 \, {\left (12 \, x^{2} \log \left (x\right )^{2} - x^{2}\right )} \log \left (\log \left (x\right )\right )^{2} + x^{2} + 8 \, {\left (4 \, x^{2} \log \left (x\right )^{3} - x^{2} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) - 2 \, x} \]

[In]

integrate(((x^2-10*x)*log(x)*log(x^2*log(x))^4+((8*x^2-40*x)*log(x)+4*x^2-20*x)*log(x^2*log(x))^3+(-2*x^2+20*x
)*log(x)*log(x^2*log(x))^2+((-8*x^2+40*x)*log(x)-4*x^2+20*x)*log(x^2*log(x))+(x^2-10*x+10)*log(x))/(x^4*log(x)
*log(x^2*log(x))^8-4*x^4*log(x)*log(x^2*log(x))^6+(6*x^4-4*x^3)*log(x)*log(x^2*log(x))^4+(-4*x^4+8*x^3)*log(x)
*log(x^2*log(x))^2+(x^4-4*x^3+4*x^2)*log(x)),x, algorithm="maxima")

[Out]

-(x - 5)/(16*x^2*log(x)^4 + 8*x^2*log(x)*log(log(x))^3 + x^2*log(log(x))^4 - 8*x^2*log(x)^2 + 2*(12*x^2*log(x)
^2 - x^2)*log(log(x))^2 + x^2 + 8*(4*x^2*log(x)^3 - x^2*log(x))*log(log(x)) - 2*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (28) = 56\).

Time = 77.69 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.54 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=-\frac {x - 5}{16 \, x^{2} \log \left (x\right )^{4} + 32 \, x^{2} \log \left (x\right )^{3} \log \left (\log \left (x\right )\right ) + 24 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x\right )\right )^{2} + 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + x^{2} \log \left (\log \left (x\right )\right )^{4} - 8 \, x^{2} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 2 \, x^{2} \log \left (\log \left (x\right )\right )^{2} + x^{2} - 2 \, x} \]

[In]

integrate(((x^2-10*x)*log(x)*log(x^2*log(x))^4+((8*x^2-40*x)*log(x)+4*x^2-20*x)*log(x^2*log(x))^3+(-2*x^2+20*x
)*log(x)*log(x^2*log(x))^2+((-8*x^2+40*x)*log(x)-4*x^2+20*x)*log(x^2*log(x))+(x^2-10*x+10)*log(x))/(x^4*log(x)
*log(x^2*log(x))^8-4*x^4*log(x)*log(x^2*log(x))^6+(6*x^4-4*x^3)*log(x)*log(x^2*log(x))^4+(-4*x^4+8*x^3)*log(x)
*log(x^2*log(x))^2+(x^4-4*x^3+4*x^2)*log(x)),x, algorithm="giac")

[Out]

-(x - 5)/(16*x^2*log(x)^4 + 32*x^2*log(x)^3*log(log(x)) + 24*x^2*log(x)^2*log(log(x))^2 + 8*x^2*log(x)*log(log
(x))^3 + x^2*log(log(x))^4 - 8*x^2*log(x)^2 - 8*x^2*log(x)*log(log(x)) - 2*x^2*log(log(x))^2 + x^2 - 2*x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\int \frac {-\ln \left (x\right )\,\left (10\,x-x^2\right )\,{\ln \left (x^2\,\ln \left (x\right )\right )}^4+\left (4\,x^2-\ln \left (x\right )\,\left (40\,x-8\,x^2\right )-20\,x\right )\,{\ln \left (x^2\,\ln \left (x\right )\right )}^3+\ln \left (x\right )\,\left (20\,x-2\,x^2\right )\,{\ln \left (x^2\,\ln \left (x\right )\right )}^2+\left (20\,x+\ln \left (x\right )\,\left (40\,x-8\,x^2\right )-4\,x^2\right )\,\ln \left (x^2\,\ln \left (x\right )\right )+\ln \left (x\right )\,\left (x^2-10\,x+10\right )}{\ln \left (x\right )\,\left (x^4-4\,x^3+4\,x^2\right )-4\,x^4\,{\ln \left (x^2\,\ln \left (x\right )\right )}^6\,\ln \left (x\right )+x^4\,{\ln \left (x^2\,\ln \left (x\right )\right )}^8\,\ln \left (x\right )+{\ln \left (x^2\,\ln \left (x\right )\right )}^2\,\ln \left (x\right )\,\left (8\,x^3-4\,x^4\right )-{\ln \left (x^2\,\ln \left (x\right )\right )}^4\,\ln \left (x\right )\,\left (4\,x^3-6\,x^4\right )} \,d x \]

[In]

int((log(x^2*log(x))*(20*x + log(x)*(40*x - 8*x^2) - 4*x^2) + log(x)*(x^2 - 10*x + 10) - log(x^2*log(x))^3*(20
*x + log(x)*(40*x - 8*x^2) - 4*x^2) - log(x^2*log(x))^4*log(x)*(10*x - x^2) + log(x^2*log(x))^2*log(x)*(20*x -
2*x^2))/(log(x)*(4*x^2 - 4*x^3 + x^4) - 4*x^4*log(x^2*log(x))^6*log(x) + x^4*log(x^2*log(x))^8*log(x) + log(x
^2*log(x))^2*log(x)*(8*x^3 - 4*x^4) - log(x^2*log(x))^4*log(x)*(4*x^3 - 6*x^4)),x)

[Out]

int((log(x^2*log(x))*(20*x + log(x)*(40*x - 8*x^2) - 4*x^2) + log(x)*(x^2 - 10*x + 10) - log(x^2*log(x))^3*(20
*x + log(x)*(40*x - 8*x^2) - 4*x^2) - log(x^2*log(x))^4*log(x)*(10*x - x^2) + log(x^2*log(x))^2*log(x)*(20*x -
2*x^2))/(log(x)*(4*x^2 - 4*x^3 + x^4) - 4*x^4*log(x^2*log(x))^6*log(x) + x^4*log(x^2*log(x))^8*log(x) + log(x
^2*log(x))^2*log(x)*(8*x^3 - 4*x^4) - log(x^2*log(x))^4*log(x)*(4*x^3 - 6*x^4)), x)