Integrand size = 210, antiderivative size = 28 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\frac {-5+x}{2 x-\left (x-x \log ^2\left (x^2 \log (x)\right )\right )^2} \] Output:
(-5+x)/(2*x-(x-x*ln(x^2*ln(x))^2)^2)
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\frac {5-x}{x \left (-2+x-2 x \log ^2\left (x^2 \log (x)\right )+x \log ^4\left (x^2 \log (x)\right )\right )} \] Input:
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]*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]
Output:
(5 - x)/(x*(-2 + x - 2*x*Log[x^2*Log[x]]^2 + x*Log[x^2*Log[x]]^4))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (x^2-10 x\right ) \log (x) \log ^4\left (x^2 \log (x)\right )+\left (4 x^2+\left (8 x^2-40 x\right ) \log (x)-20 x\right ) \log ^3\left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^2+\left (40 x-8 x^2\right ) \log (x)+20 x\right ) \log \left (x^2 \log (x)\right )+\left (x^2-10 x+10\right ) \log (x)}{x^4 \log (x) \log ^8\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+\left (6 x^4-4 x^3\right ) \log (x) \log ^4\left (x^2 \log (x)\right )+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (x^4-4 x^3+4 x^2\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (x^2-10 x\right ) \log (x) \log ^4\left (x^2 \log (x)\right )+\left (4 x^2+\left (8 x^2-40 x\right ) \log (x)-20 x\right ) \log ^3\left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^2+\left (40 x-8 x^2\right ) \log (x)+20 x\right ) \log \left (x^2 \log (x)\right )+\left (x^2-10 x+10\right ) \log (x)}{x^2 \log (x) \left (-x \log ^4\left (x^2 \log (x)\right )+2 x \log ^2\left (x^2 \log (x)\right )-x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x-10}{x^2 \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )}+\frac {2 (x-5) \left (2 x \log ^3\left (x^2 \log (x)\right )+4 x \log (x) \log ^3\left (x^2 \log (x)\right )-2 x \log \left (x^2 \log (x)\right )-4 x \log (x) \log \left (x^2 \log (x)\right )+\log (x)\right )}{x^2 \log (x) \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -10 \int \frac {1}{x^2 \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx+2 \int \frac {1}{x \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx-8 \int \frac {\log \left (x^2 \log (x)\right )}{\left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx+40 \int \frac {\log \left (x^2 \log (x)\right )}{x \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx-4 \int \frac {\log \left (x^2 \log (x)\right )}{\log (x) \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx+20 \int \frac {\log \left (x^2 \log (x)\right )}{x \log (x) \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx-10 \int \frac {1}{x^2 \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )}dx+\int \frac {1}{x \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )}dx+8 \int \frac {\log ^3\left (x^2 \log (x)\right )}{\left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx-40 \int \frac {\log ^3\left (x^2 \log (x)\right )}{x \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx+4 \int \frac {\log ^3\left (x^2 \log (x)\right )}{\log (x) \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx-20 \int \frac {\log ^3\left (x^2 \log (x)\right )}{x \log (x) \left (x \log ^4\left (x^2 \log (x)\right )-2 x \log ^2\left (x^2 \log (x)\right )+x-2\right )^2}dx\) |
Input:
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*L og[x]]^4)/((4*x^2 - 4*x^3 + x^4)*Log[x] + (8*x^3 - 4*x^4)*Log[x]*Log[x^2*L og[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]
Output:
$Aborted
Timed out.
\[\int \frac {\left (x^{2}-10 x \right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{4}+\left (\left (8 x^{2}-40 x \right ) \ln \left (x \right )+4 x^{2}-20 x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{3}+\left (-2 x^{2}+20 x \right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{2}+\left (\left (-8 x^{2}+40 x \right ) \ln \left (x \right )-4 x^{2}+20 x \right ) \ln \left (x^{2} \ln \left (x \right )\right )+\left (x^{2}-10 x +10\right ) \ln \left (x \right )}{x^{4} \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{8}-4 x^{4} \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{6}+\left (6 x^{4}-4 x^{3}\right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{4}+\left (-4 x^{4}+8 x^{3}\right ) \ln \left (x \right ) \ln \left (x^{2} \ln \left (x \right )\right )^{2}+\left (x^{4}-4 x^{3}+4 x^{2}\right ) \ln \left (x \right )}d x\]
Input:
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)
Output:
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)
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=-\frac {x - 5}{x^{2} \log \left (x^{2} \log \left (x\right )\right )^{4} - 2 \, x^{2} \log \left (x^{2} \log \left (x\right )\right )^{2} + x^{2} - 2 \, x} \] Input:
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)*l og(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")
Output:
-(x - 5)/(x^2*log(x^2*log(x))^4 - 2*x^2*log(x^2*log(x))^2 + x^2 - 2*x)
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\frac {5 - x}{x^{2} \log {\left (x^{2} \log {\left (x \right )} \right )}^{4} - 2 x^{2} \log {\left (x^{2} \log {\left (x \right )} \right )}^{2} + x^{2} - 2 x} \] Input:
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)
Output:
(5 - x)/(x**2*log(x**2*log(x))**4 - 2*x**2*log(x**2*log(x))**2 + x**2 - 2* x)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (28) = 56\).
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=-\frac {x - 5}{16 \, x^{2} \log \left (x\right )^{4} + 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + x^{2} \log \left (\log \left (x\right )\right )^{4} - 8 \, x^{2} \log \left (x\right )^{2} + 2 \, {\left (12 \, x^{2} \log \left (x\right )^{2} - x^{2}\right )} \log \left (\log \left (x\right )\right )^{2} + x^{2} + 8 \, {\left (4 \, x^{2} \log \left (x\right )^{3} - x^{2} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) - 2 \, x} \] Input:
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)*l og(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")
Output:
-(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)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (28) = 56\).
Time = 75.94 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.54 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=-\frac {x - 5}{16 \, x^{2} \log \left (x\right )^{4} + 32 \, x^{2} \log \left (x\right )^{3} \log \left (\log \left (x\right )\right ) + 24 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x\right )\right )^{2} + 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{3} + x^{2} \log \left (\log \left (x\right )\right )^{4} - 8 \, x^{2} \log \left (x\right )^{2} - 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 2 \, x^{2} \log \left (\log \left (x\right )\right )^{2} + x^{2} - 2 \, x} \] Input:
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)*l og(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")
Output:
-(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)
Timed out. \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\int \frac {-\ln \left (x\right )\,\left (10\,x-x^2\right )\,{\ln \left (x^2\,\ln \left (x\right )\right )}^4+\left (4\,x^2-\ln \left (x\right )\,\left (40\,x-8\,x^2\right )-20\,x\right )\,{\ln \left (x^2\,\ln \left (x\right )\right )}^3+\ln \left (x\right )\,\left (20\,x-2\,x^2\right )\,{\ln \left (x^2\,\ln \left (x\right )\right )}^2+\left (20\,x+\ln \left (x\right )\,\left (40\,x-8\,x^2\right )-4\,x^2\right )\,\ln \left (x^2\,\ln \left (x\right )\right )+\ln \left (x\right )\,\left (x^2-10\,x+10\right )}{\ln \left (x\right )\,\left (x^4-4\,x^3+4\,x^2\right )-4\,x^4\,{\ln \left (x^2\,\ln \left (x\right )\right )}^6\,\ln \left (x\right )+x^4\,{\ln \left (x^2\,\ln \left (x\right )\right )}^8\,\ln \left (x\right )+{\ln \left (x^2\,\ln \left (x\right )\right )}^2\,\ln \left (x\right )\,\left (8\,x^3-4\,x^4\right )-{\ln \left (x^2\,\ln \left (x\right )\right )}^4\,\ln \left (x\right )\,\left (4\,x^3-6\,x^4\right )} \,d x \] Input:
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) - lo g(x^2*log(x))^4*log(x)*(4*x^3 - 6*x^4)),x)
Output:
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) - lo g(x^2*log(x))^4*log(x)*(4*x^3 - 6*x^4)), x)
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (10-10 x+x^2\right ) \log (x)+\left (20 x-4 x^2+\left (40 x-8 x^2\right ) \log (x)\right ) \log \left (x^2 \log (x)\right )+\left (20 x-2 x^2\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-20 x+4 x^2+\left (-40 x+8 x^2\right ) \log (x)\right ) \log ^3\left (x^2 \log (x)\right )+\left (-10 x+x^2\right ) \log (x) \log ^4\left (x^2 \log (x)\right )}{\left (4 x^2-4 x^3+x^4\right ) \log (x)+\left (8 x^3-4 x^4\right ) \log (x) \log ^2\left (x^2 \log (x)\right )+\left (-4 x^3+6 x^4\right ) \log (x) \log ^4\left (x^2 \log (x)\right )-4 x^4 \log (x) \log ^6\left (x^2 \log (x)\right )+x^4 \log (x) \log ^8\left (x^2 \log (x)\right )} \, dx=\frac {-x +5}{x \left (\mathrm {log}\left (\mathrm {log}\left (x \right ) x^{2}\right )^{4} x -2 \mathrm {log}\left (\mathrm {log}\left (x \right ) x^{2}\right )^{2} x +x -2\right )} \] Input:
int(((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)*lo g(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 )
Output:
( - x + 5)/(x*(log(log(x)*x**2)**4*x - 2*log(log(x)*x**2)**2*x + x - 2))