\(\int \frac {-2 x^2-2 x \log (x)+(-2 x-4 x^2-2 x \log (x)) \log (25 x)+(2 x^2+2 x \log (x)) \log (25 x) \log (x+\log (x))+(2 x^2+2 x \log (x)) \log (25 x) \log (\log (25 x))}{(-x-3 x^2-3 x^3-x^4+(-1-3 x-3 x^2-x^3) \log (x)) \log (25 x)+(3 x+6 x^2+3 x^3+(3+6 x+3 x^2) \log (x)) \log (25 x) \log (x+\log (x))+(-3 x-3 x^2+(-3-3 x) \log (x)) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+((3 x+6 x^2+3 x^3+(3+6 x+3 x^2) \log (x)) \log (25 x)+(-6 x-6 x^2+(-6-6 x) \log (x)) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))) \log (\log (25 x))+((-3 x-3 x^2+(-3-3 x) \log (x)) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx\) [1072]

Optimal result

Integrand size = 351, antiderivative size = 21 \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\frac {x^2}{(-1-x+\log (x+\log (x))+\log (\log (25 x)))^2} \] Output:

x^2/(ln(x+ln(x))+ln(ln(25*x))-1-x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\frac {x^2}{(-1-x+\log (x+\log (x))+\log (\log (25 x)))^2} \] Input:

Integrate[(-2*x^2 - 2*x*Log[x] + (-2*x - 4*x^2 - 2*x*Log[x])*Log[25*x] + ( 
2*x^2 + 2*x*Log[x])*Log[25*x]*Log[x + Log[x]] + (2*x^2 + 2*x*Log[x])*Log[2 
5*x]*Log[Log[25*x]])/((-x - 3*x^2 - 3*x^3 - x^4 + (-1 - 3*x - 3*x^2 - x^3) 
*Log[x])*Log[25*x] + (3*x + 6*x^2 + 3*x^3 + (3 + 6*x + 3*x^2)*Log[x])*Log[ 
25*x]*Log[x + Log[x]] + (-3*x - 3*x^2 + (-3 - 3*x)*Log[x])*Log[25*x]*Log[x 
 + Log[x]]^2 + (x + Log[x])*Log[25*x]*Log[x + Log[x]]^3 + ((3*x + 6*x^2 + 
3*x^3 + (3 + 6*x + 3*x^2)*Log[x])*Log[25*x] + (-6*x - 6*x^2 + (-6 - 6*x)*L 
og[x])*Log[25*x]*Log[x + Log[x]] + (3*x + 3*Log[x])*Log[25*x]*Log[x + Log[ 
x]]^2)*Log[Log[25*x]] + ((-3*x - 3*x^2 + (-3 - 3*x)*Log[x])*Log[25*x] + (3 
*x + 3*Log[x])*Log[25*x]*Log[x + Log[x]])*Log[Log[25*x]]^2 + (x + Log[x])* 
Log[25*x]*Log[Log[25*x]]^3),x]
 

Output:

x^2/(-1 - x + Log[x + Log[x]] + Log[Log[25*x]])^2
 

Rubi [F]

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.

\(\displaystyle \int \frac {-2 x^2+\left (-4 x^2-2 x-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))-2 x \log (x)}{\left (-3 x^2-3 x+(-3 x-3) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+\left (\left (-3 x^2-3 x+(-3 x-3) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (x+\log (x)) \log (25 x)\right ) \log ^2(\log (25 x))+\left (\left (-6 x^2-6 x+(-6 x-6) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (3 x^3+6 x^2+\left (3 x^2+6 x+3\right ) \log (x)+3 x\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (3 x^3+6 x^2+\left (3 x^2+6 x+3\right ) \log (x)+3 x\right ) \log (25 x) \log (x+\log (x))+\left (-x^4-3 x^3-3 x^2+\left (-x^3-3 x^2-3 x-1\right ) \log (x)-x\right ) \log (25 x)+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 x (x-\log (25 x) (-2 x+x \log (x+\log (x))+x \log (\log (25 x))-1)-\log (x) (\log (25 x) (\log (x+\log (x))+\log (\log (25 x))-1)-1))}{(x+\log (x)) \log (25 x) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {x (x+\log (x) (\log (25 x) (-\log (x+\log (x))-\log (\log (25 x))+1)+1)+\log (25 x) (-\log (x+\log (x)) x-\log (\log (25 x)) x+2 x+1))}{(x+\log (x)) \log (25 x) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {x}{(x-\log (x+\log (x))-\log (\log (25 x))+1)^2}-\frac {x \left (\log (25 x) x^2+\log (x) \log (25 x) x-\log (25 x) x-x-\log (x)-\log (25 x)\right )}{(x+\log (x)) \log (25 x) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\int \frac {x^3}{(x+\log (x)) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx+\int \frac {x^2}{(x+\log (x)) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx-\int \frac {x^2 \log (x)}{(x+\log (x)) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx+\int \frac {x^2}{(x+\log (x)) \log (25 x) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx+\int \frac {x}{(x+\log (x)) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx+\int \frac {x \log (x)}{(x+\log (x)) \log (25 x) (x-\log (x+\log (x))-\log (\log (25 x))+1)^3}dx+\int \frac {x}{(x-\log (x+\log (x))-\log (\log (25 x))+1)^2}dx\right )\)

Input:

Int[(-2*x^2 - 2*x*Log[x] + (-2*x - 4*x^2 - 2*x*Log[x])*Log[25*x] + (2*x^2 
+ 2*x*Log[x])*Log[25*x]*Log[x + Log[x]] + (2*x^2 + 2*x*Log[x])*Log[25*x]*L 
og[Log[25*x]])/((-x - 3*x^2 - 3*x^3 - x^4 + (-1 - 3*x - 3*x^2 - x^3)*Log[x 
])*Log[25*x] + (3*x + 6*x^2 + 3*x^3 + (3 + 6*x + 3*x^2)*Log[x])*Log[25*x]* 
Log[x + Log[x]] + (-3*x - 3*x^2 + (-3 - 3*x)*Log[x])*Log[25*x]*Log[x + Log 
[x]]^2 + (x + Log[x])*Log[25*x]*Log[x + Log[x]]^3 + ((3*x + 6*x^2 + 3*x^3 
+ (3 + 6*x + 3*x^2)*Log[x])*Log[25*x] + (-6*x - 6*x^2 + (-6 - 6*x)*Log[x]) 
*Log[25*x]*Log[x + Log[x]] + (3*x + 3*Log[x])*Log[25*x]*Log[x + Log[x]]^2) 
*Log[Log[25*x]] + ((-3*x - 3*x^2 + (-3 - 3*x)*Log[x])*Log[25*x] + (3*x + 3 
*Log[x])*Log[25*x]*Log[x + Log[x]])*Log[Log[25*x]]^2 + (x + Log[x])*Log[25 
*x]*Log[Log[25*x]]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29

\[\frac {x^{2}}{\left (x -\ln \left (2 \ln \left (5\right )+\ln \left (x \right )\right )-\ln \left (x +\ln \left (x \right )\right )+1\right )^{2}}\]

Input:

int(((2*x*ln(x)+2*x^2)*ln(25*x)*ln(ln(25*x))+(2*x*ln(x)+2*x^2)*ln(25*x)*ln 
(x+ln(x))+(-2*x*ln(x)-4*x^2-2*x)*ln(25*x)-2*x*ln(x)-2*x^2)/((x+ln(x))*ln(2 
5*x)*ln(ln(25*x))^3+((3*x+3*ln(x))*ln(25*x)*ln(x+ln(x))+((-3*x-3)*ln(x)-3* 
x^2-3*x)*ln(25*x))*ln(ln(25*x))^2+((3*x+3*ln(x))*ln(25*x)*ln(x+ln(x))^2+(( 
-6*x-6)*ln(x)-6*x^2-6*x)*ln(25*x)*ln(x+ln(x))+((3*x^2+6*x+3)*ln(x)+3*x^3+6 
*x^2+3*x)*ln(25*x))*ln(ln(25*x))+(x+ln(x))*ln(25*x)*ln(x+ln(x))^3+((-3*x-3 
)*ln(x)-3*x^2-3*x)*ln(25*x)*ln(x+ln(x))^2+((3*x^2+6*x+3)*ln(x)+3*x^3+6*x^2 
+3*x)*ln(25*x)*ln(x+ln(x))+((-x^3-3*x^2-3*x-1)*ln(x)-x^4-3*x^3-3*x^2-x)*ln 
(25*x)),x)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\frac {x^{2}}{x^{2} - 2 \, {\left (x + 1\right )} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} - 2 \, {\left (x - \log \left (x + \log \left (x\right )\right ) + 1\right )} \log \left (2 \, \log \left (5\right ) + \log \left (x\right )\right ) + \log \left (2 \, \log \left (5\right ) + \log \left (x\right )\right )^{2} + 2 \, x + 1} \] Input:

integrate(((2*x*log(x)+2*x^2)*log(25*x)*log(log(25*x))+(2*x*log(x)+2*x^2)* 
log(25*x)*log(x+log(x))+(-2*x*log(x)-4*x^2-2*x)*log(25*x)-2*x*log(x)-2*x^2 
)/((x+log(x))*log(25*x)*log(log(25*x))^3+((3*x+3*log(x))*log(25*x)*log(x+l 
og(x))+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x))*log(log(25*x))^2+((3*x+3*log 
(x))*log(25*x)*log(x+log(x))^2+((-6*x-6)*log(x)-6*x^2-6*x)*log(25*x)*log(x 
+log(x))+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x))*log(log(25*x))+ 
(x+log(x))*log(25*x)*log(x+log(x))^3+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x) 
*log(x+log(x))^2+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x)*log(x+lo 
g(x))+((-x^3-3*x^2-3*x-1)*log(x)-x^4-3*x^3-3*x^2-x)*log(25*x)),x, algorith 
m="fricas")
 

Output:

x^2/(x^2 - 2*(x + 1)*log(x + log(x)) + log(x + log(x))^2 - 2*(x - log(x + 
log(x)) + 1)*log(2*log(5) + log(x)) + log(2*log(5) + log(x))^2 + 2*x + 1)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).

Time = 0.58 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.33 \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\frac {x^{2}}{x^{2} - 2 x \log {\left (x + \log {\left (x \right )} \right )} + 2 x + \left (- 2 x + 2 \log {\left (x + \log {\left (x \right )} \right )} - 2\right ) \log {\left (\log {\left (x \right )} + \log {\left (25 \right )} \right )} + \log {\left (x + \log {\left (x \right )} \right )}^{2} - 2 \log {\left (x + \log {\left (x \right )} \right )} + \log {\left (\log {\left (x \right )} + \log {\left (25 \right )} \right )}^{2} + 1} \] Input:

integrate(((2*x*ln(x)+2*x**2)*ln(25*x)*ln(ln(25*x))+(2*x*ln(x)+2*x**2)*ln( 
25*x)*ln(x+ln(x))+(-2*x*ln(x)-4*x**2-2*x)*ln(25*x)-2*x*ln(x)-2*x**2)/((x+l 
n(x))*ln(25*x)*ln(ln(25*x))**3+((3*x+3*ln(x))*ln(25*x)*ln(x+ln(x))+((-3*x- 
3)*ln(x)-3*x**2-3*x)*ln(25*x))*ln(ln(25*x))**2+((3*x+3*ln(x))*ln(25*x)*ln( 
x+ln(x))**2+((-6*x-6)*ln(x)-6*x**2-6*x)*ln(25*x)*ln(x+ln(x))+((3*x**2+6*x+ 
3)*ln(x)+3*x**3+6*x**2+3*x)*ln(25*x))*ln(ln(25*x))+(x+ln(x))*ln(25*x)*ln(x 
+ln(x))**3+((-3*x-3)*ln(x)-3*x**2-3*x)*ln(25*x)*ln(x+ln(x))**2+((3*x**2+6* 
x+3)*ln(x)+3*x**3+6*x**2+3*x)*ln(25*x)*ln(x+ln(x))+((-x**3-3*x**2-3*x-1)*l 
n(x)-x**4-3*x**3-3*x**2-x)*ln(25*x)),x)
 

Output:

x**2/(x**2 - 2*x*log(x + log(x)) + 2*x + (-2*x + 2*log(x + log(x)) - 2)*lo 
g(log(x) + log(25)) + log(x + log(x))**2 - 2*log(x + log(x)) + log(log(x) 
+ log(25))**2 + 1)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).

Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\frac {x^{2}}{x^{2} - 2 \, {\left (x + 1\right )} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} - 2 \, {\left (x - \log \left (x + \log \left (x\right )\right ) + 1\right )} \log \left (2 \, \log \left (5\right ) + \log \left (x\right )\right ) + \log \left (2 \, \log \left (5\right ) + \log \left (x\right )\right )^{2} + 2 \, x + 1} \] Input:

integrate(((2*x*log(x)+2*x^2)*log(25*x)*log(log(25*x))+(2*x*log(x)+2*x^2)* 
log(25*x)*log(x+log(x))+(-2*x*log(x)-4*x^2-2*x)*log(25*x)-2*x*log(x)-2*x^2 
)/((x+log(x))*log(25*x)*log(log(25*x))^3+((3*x+3*log(x))*log(25*x)*log(x+l 
og(x))+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x))*log(log(25*x))^2+((3*x+3*log 
(x))*log(25*x)*log(x+log(x))^2+((-6*x-6)*log(x)-6*x^2-6*x)*log(25*x)*log(x 
+log(x))+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x))*log(log(25*x))+ 
(x+log(x))*log(25*x)*log(x+log(x))^3+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x) 
*log(x+log(x))^2+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x)*log(x+lo 
g(x))+((-x^3-3*x^2-3*x-1)*log(x)-x^4-3*x^3-3*x^2-x)*log(25*x)),x, algorith 
m="maxima")
 

Output:

x^2/(x^2 - 2*(x + 1)*log(x + log(x)) + log(x + log(x))^2 - 2*(x - log(x + 
log(x)) + 1)*log(2*log(5) + log(x)) + log(2*log(5) + log(x))^2 + 2*x + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (23) = 46\).

Time = 0.53 (sec) , antiderivative size = 900, normalized size of antiderivative = 42.86 \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\text {Too large to display} \] Input:

integrate(((2*x*log(x)+2*x^2)*log(25*x)*log(log(25*x))+(2*x*log(x)+2*x^2)* 
log(25*x)*log(x+log(x))+(-2*x*log(x)-4*x^2-2*x)*log(25*x)-2*x*log(x)-2*x^2 
)/((x+log(x))*log(25*x)*log(log(25*x))^3+((3*x+3*log(x))*log(25*x)*log(x+l 
og(x))+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x))*log(log(25*x))^2+((3*x+3*log 
(x))*log(25*x)*log(x+log(x))^2+((-6*x-6)*log(x)-6*x^2-6*x)*log(25*x)*log(x 
+log(x))+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x))*log(log(25*x))+ 
(x+log(x))*log(25*x)*log(x+log(x))^3+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x) 
*log(x+log(x))^2+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x)*log(x+lo 
g(x))+((-x^3-3*x^2-3*x-1)*log(x)-x^4-3*x^3-3*x^2-x)*log(25*x)),x, algorith 
m="giac")
 

Output:

(2*x^4*log(5) + x^4*log(x) + 2*x^3*log(5)*log(x) + x^3*log(x)^2 - 2*x^3*lo 
g(5) - x^3*log(x) - x^3 - 2*x^2*log(5) - 2*x^2*log(x))/(2*x^4*log(5) - 4*x 
^3*log(5)*log(x + log(x)) + 2*x^2*log(5)*log(x + log(x))^2 + x^4*log(x) + 
2*x^3*log(5)*log(x) - 2*x^3*log(x + log(x))*log(x) - 4*x^2*log(5)*log(x + 
log(x))*log(x) + x^2*log(x + log(x))^2*log(x) + 2*x*log(5)*log(x + log(x)) 
^2*log(x) + x^3*log(x)^2 - 2*x^2*log(x + log(x))*log(x)^2 + x*log(x + log( 
x))^2*log(x)^2 - 4*x^3*log(5)*log(2*log(5) + log(x)) + 4*x^2*log(5)*log(x 
+ log(x))*log(2*log(5) + log(x)) - 2*x^3*log(x)*log(2*log(5) + log(x)) - 4 
*x^2*log(5)*log(x)*log(2*log(5) + log(x)) + 2*x^2*log(x + log(x))*log(x)*l 
og(2*log(5) + log(x)) + 4*x*log(5)*log(x + log(x))*log(x)*log(2*log(5) + l 
og(x)) - 2*x^2*log(x)^2*log(2*log(5) + log(x)) + 2*x*log(x + log(x))*log(x 
)^2*log(2*log(5) + log(x)) + 2*x^2*log(5)*log(2*log(5) + log(x))^2 + x^2*l 
og(x)*log(2*log(5) + log(x))^2 + 2*x*log(5)*log(x)*log(2*log(5) + log(x))^ 
2 + x*log(x)^2*log(2*log(5) + log(x))^2 + 2*x^3*log(5) - 2*x*log(5)*log(x 
+ log(x))^2 + x^3*log(x) + 4*x^2*log(5)*log(x) - 4*x*log(5)*log(x + log(x) 
)*log(x) - x*log(x + log(x))^2*log(x) + 2*x^2*log(x)^2 - 2*x*log(x + log(x 
))*log(x)^2 - 4*x*log(5)*log(x + log(x))*log(2*log(5) + log(x)) - 4*x*log( 
5)*log(x)*log(2*log(5) + log(x)) - 2*x*log(x + log(x))*log(x)*log(2*log(5) 
 + log(x)) - 2*x*log(x)^2*log(2*log(5) + log(x)) - 2*x*log(5)*log(2*log(5) 
 + log(x))^2 - x*log(x)*log(2*log(5) + log(x))^2 - x^3 - 4*x^2*log(5) +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\int -\frac {2\,x\,\ln \left (x\right )+\ln \left (25\,x\right )\,\left (2\,x+2\,x\,\ln \left (x\right )+4\,x^2\right )+2\,x^2-\ln \left (x+\ln \left (x\right )\right )\,\ln \left (25\,x\right )\,\left (2\,x\,\ln \left (x\right )+2\,x^2\right )-\ln \left (25\,x\right )\,\ln \left (\ln \left (25\,x\right )\right )\,\left (2\,x\,\ln \left (x\right )+2\,x^2\right )}{\ln \left (\ln \left (25\,x\right )\right )\,\left (\ln \left (25\,x\right )\,\left (3\,x+3\,\ln \left (x\right )\right )\,{\ln \left (x+\ln \left (x\right )\right )}^2-\ln \left (25\,x\right )\,\left (6\,x+\ln \left (x\right )\,\left (6\,x+6\right )+6\,x^2\right )\,\ln \left (x+\ln \left (x\right )\right )+\ln \left (25\,x\right )\,\left (3\,x+\ln \left (x\right )\,\left (3\,x^2+6\,x+3\right )+6\,x^2+3\,x^3\right )\right )-{\ln \left (\ln \left (25\,x\right )\right )}^2\,\left (\ln \left (25\,x\right )\,\left (3\,x+\ln \left (x\right )\,\left (3\,x+3\right )+3\,x^2\right )-\ln \left (x+\ln \left (x\right )\right )\,\ln \left (25\,x\right )\,\left (3\,x+3\,\ln \left (x\right )\right )\right )-\ln \left (25\,x\right )\,\left (x+\ln \left (x\right )\,\left (x^3+3\,x^2+3\,x+1\right )+3\,x^2+3\,x^3+x^4\right )-{\ln \left (x+\ln \left (x\right )\right )}^2\,\ln \left (25\,x\right )\,\left (3\,x+\ln \left (x\right )\,\left (3\,x+3\right )+3\,x^2\right )+\ln \left (x+\ln \left (x\right )\right )\,\ln \left (25\,x\right )\,\left (3\,x+\ln \left (x\right )\,\left (3\,x^2+6\,x+3\right )+6\,x^2+3\,x^3\right )+\ln \left (25\,x\right )\,{\ln \left (\ln \left (25\,x\right )\right )}^3\,\left (x+\ln \left (x\right )\right )+{\ln \left (x+\ln \left (x\right )\right )}^3\,\ln \left (25\,x\right )\,\left (x+\ln \left (x\right )\right )} \,d x \] Input:

int(-(2*x*log(x) + log(25*x)*(2*x + 2*x*log(x) + 4*x^2) + 2*x^2 - log(x + 
log(x))*log(25*x)*(2*x*log(x) + 2*x^2) - log(25*x)*log(log(25*x))*(2*x*log 
(x) + 2*x^2))/(log(log(25*x))*(log(25*x)*(3*x + log(x)*(6*x + 3*x^2 + 3) + 
 6*x^2 + 3*x^3) + log(x + log(x))^2*log(25*x)*(3*x + 3*log(x)) - log(x + l 
og(x))*log(25*x)*(6*x + log(x)*(6*x + 6) + 6*x^2)) - log(log(25*x))^2*(log 
(25*x)*(3*x + log(x)*(3*x + 3) + 3*x^2) - log(x + log(x))*log(25*x)*(3*x + 
 3*log(x))) - log(25*x)*(x + log(x)*(3*x + 3*x^2 + x^3 + 1) + 3*x^2 + 3*x^ 
3 + x^4) - log(x + log(x))^2*log(25*x)*(3*x + log(x)*(3*x + 3) + 3*x^2) + 
log(x + log(x))*log(25*x)*(3*x + log(x)*(6*x + 3*x^2 + 3) + 6*x^2 + 3*x^3) 
 + log(25*x)*log(log(25*x))^3*(x + log(x)) + log(x + log(x))^3*log(25*x)*( 
x + log(x))),x)
 

Output:

int(-(2*x*log(x) + log(25*x)*(2*x + 2*x*log(x) + 4*x^2) + 2*x^2 - log(x + 
log(x))*log(25*x)*(2*x*log(x) + 2*x^2) - log(25*x)*log(log(25*x))*(2*x*log 
(x) + 2*x^2))/(log(log(25*x))*(log(25*x)*(3*x + log(x)*(6*x + 3*x^2 + 3) + 
 6*x^2 + 3*x^3) + log(x + log(x))^2*log(25*x)*(3*x + 3*log(x)) - log(x + l 
og(x))*log(25*x)*(6*x + log(x)*(6*x + 6) + 6*x^2)) - log(log(25*x))^2*(log 
(25*x)*(3*x + log(x)*(3*x + 3) + 3*x^2) - log(x + log(x))*log(25*x)*(3*x + 
 3*log(x))) - log(25*x)*(x + log(x)*(3*x + 3*x^2 + x^3 + 1) + 3*x^2 + 3*x^ 
3 + x^4) - log(x + log(x))^2*log(25*x)*(3*x + log(x)*(3*x + 3) + 3*x^2) + 
log(x + log(x))*log(25*x)*(3*x + log(x)*(6*x + 3*x^2 + 3) + 6*x^2 + 3*x^3) 
 + log(25*x)*log(log(25*x))^3*(x + log(x)) + log(x + log(x))^3*log(25*x)*( 
x + log(x))), x)
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.33 \[ \int \frac {-2 x^2-2 x \log (x)+\left (-2 x-4 x^2-2 x \log (x)\right ) \log (25 x)+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (2 x^2+2 x \log (x)\right ) \log (25 x) \log (\log (25 x))}{\left (-x-3 x^2-3 x^3-x^4+\left (-1-3 x-3 x^2-x^3\right ) \log (x)\right ) \log (25 x)+\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x) \log (x+\log (x))+\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x) \log ^2(x+\log (x))+(x+\log (x)) \log (25 x) \log ^3(x+\log (x))+\left (\left (3 x+6 x^2+3 x^3+\left (3+6 x+3 x^2\right ) \log (x)\right ) \log (25 x)+\left (-6 x-6 x^2+(-6-6 x) \log (x)\right ) \log (25 x) \log (x+\log (x))+(3 x+3 \log (x)) \log (25 x) \log ^2(x+\log (x))\right ) \log (\log (25 x))+\left (\left (-3 x-3 x^2+(-3-3 x) \log (x)\right ) \log (25 x)+(3 x+3 \log (x)) \log (25 x) \log (x+\log (x))\right ) \log ^2(\log (25 x))+(x+\log (x)) \log (25 x) \log ^3(\log (25 x))} \, dx=\frac {x^{2}}{\mathrm {log}\left (\mathrm {log}\left (25 x \right )\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (25 x \right )\right ) \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )-2 \,\mathrm {log}\left (\mathrm {log}\left (25 x \right )\right ) x -2 \,\mathrm {log}\left (\mathrm {log}\left (25 x \right )\right )+\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right ) x -2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )+x^{2}+2 x +1} \] Input:

int(((2*x*log(x)+2*x^2)*log(25*x)*log(log(25*x))+(2*x*log(x)+2*x^2)*log(25 
*x)*log(x+log(x))+(-2*x*log(x)-4*x^2-2*x)*log(25*x)-2*x*log(x)-2*x^2)/((x+ 
log(x))*log(25*x)*log(log(25*x))^3+((3*x+3*log(x))*log(25*x)*log(x+log(x)) 
+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x))*log(log(25*x))^2+((3*x+3*log(x))*l 
og(25*x)*log(x+log(x))^2+((-6*x-6)*log(x)-6*x^2-6*x)*log(25*x)*log(x+log(x 
))+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x))*log(log(25*x))+(x+log 
(x))*log(25*x)*log(x+log(x))^3+((-3*x-3)*log(x)-3*x^2-3*x)*log(25*x)*log(x 
+log(x))^2+((3*x^2+6*x+3)*log(x)+3*x^3+6*x^2+3*x)*log(25*x)*log(x+log(x))+ 
((-x^3-3*x^2-3*x-1)*log(x)-x^4-3*x^3-3*x^2-x)*log(25*x)),x)
 

Output:

x**2/(log(log(25*x))**2 + 2*log(log(25*x))*log(log(x) + x) - 2*log(log(25* 
x))*x - 2*log(log(25*x)) + log(log(x) + x)**2 - 2*log(log(x) + x)*x - 2*lo 
g(log(x) + x) + x**2 + 2*x + 1)