\(\int \frac {100+40 x+4 x^2+(-50-40 x-6 x^2) \log (x)+(10 x+2 x^2) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+((-100-20 x) \log (x)-20 \log ^3(x)) \log (-2+\log (x))+(-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx\) [1331]

Optimal result

Integrand size = 151, antiderivative size = 29 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\left (\log (x)+\frac {x+\frac {5 \left (x-x \log ^2(-2+\log (x))\right )}{x}}{\log (x)}\right )^2 \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=2 \left (x+\frac {\log ^2(x)}{2}-5 \log ^2(-2+\log (x))+\frac {\left (5+x-5 \log ^2(-2+\log (x))\right )^2}{2 \log ^2(x)}\right ) \] Input:

Integrate[(100 + 40*x + 4*x^2 + (-50 - 40*x - 6*x^2)*Log[x] + (10*x + 2*x^ 
2)*Log[x]^2 - 4*x*Log[x]^3 + (-4 + 2*x)*Log[x]^4 + 2*Log[x]^5 + ((-100 - 2 
0*x)*Log[x] - 20*Log[x]^3)*Log[-2 + Log[x]] + (-200 - 40*x + (100 + 40*x)* 
Log[x] - 10*x*Log[x]^2)*Log[-2 + Log[x]]^2 + 100*Log[x]*Log[-2 + Log[x]]^3 
 + (100 - 50*Log[x])*Log[-2 + Log[x]]^4)/(-2*x*Log[x]^3 + x*Log[x]^4),x]
 

Output:

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

\(\displaystyle \int \frac {4 x^2+\left (2 x^2+10 x\right ) \log ^2(x)+\left (-6 x^2-40 x-50\right ) \log (x)+40 x+2 \log ^5(x)+(2 x-4) \log ^4(x)+(100-50 \log (x)) \log ^4(\log (x)-2)-4 x \log ^3(x)+100 \log ^3(\log (x)-2) \log (x)+\left ((-20 x-100) \log (x)-20 \log ^3(x)\right ) \log (\log (x)-2)+\left (-40 x-10 x \log ^2(x)+(40 x+100) \log (x)-200\right ) \log ^2(\log (x)-2)+100}{x \log ^4(x)-2 x \log ^3(x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-4 x^2-\left (2 x^2+10 x\right ) \log ^2(x)-\left (-6 x^2-40 x-50\right ) \log (x)-40 x-2 \log ^5(x)-(2 x-4) \log ^4(x)-(100-50 \log (x)) \log ^4(\log (x)-2)+4 x \log ^3(x)-100 \log ^3(\log (x)-2) \log (x)-\left ((-20 x-100) \log (x)-20 \log ^3(x)\right ) \log (\log (x)-2)-\left (-40 x-10 x \log ^2(x)+(40 x+100) \log (x)-200\right ) \log ^2(\log (x)-2)-100}{x (2-\log (x)) \log ^3(x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 x}{(\log (x)-2) \log ^3(x)}+\frac {100}{x (\log (x)-2) \log ^3(x)}+\frac {40}{(\log (x)-2) \log ^3(x)}-\frac {20 \left (x+\log ^2(x)+5\right ) \log (\log (x)-2)}{x (\log (x)-2) \log ^2(x)}+\frac {2 \log ^2(x)}{x (\log (x)-2)}-\frac {2 (x+5) (3 x+5)}{x (\log (x)-2) \log ^2(x)}-\frac {50 \log ^4(\log (x)-2)}{x \log ^3(x)}+\frac {100 \log ^3(\log (x)-2)}{x (\log (x)-2) \log ^2(x)}-\frac {10 (-2 x+x \log (x)-10) \log ^2(\log (x)-2)}{x \log ^3(x)}+\frac {2 (x-2) \log (x)}{x (\log (x)-2)}-\frac {4}{\log (x)-2}+\frac {2 (x+5)}{(\log (x)-2) \log (x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {-3 x^2-20 x-25}{x \log ^2(x)}dx-\frac {1}{2} \int \frac {3 x^2+20 x+25}{x (\log (x)-2)}dx-\frac {1}{2} \int \frac {-3 x^2-20 x-25}{x \log (x)}dx+4 \int \frac {x}{(\log (x)-2) \log ^3(x)}dx-20 \int \frac {\log (\log (x)-2)}{(\log (x)-2) \log ^2(x)}dx-10 \int \frac {\log ^2(\log (x)-2)}{\log ^2(x)}dx+20 \int \frac {\log ^2(\log (x)-2)}{\log ^3(x)}dx+e^4 \operatorname {ExpIntegralEi}(-2 (2-\log (x)))+10 e^2 \operatorname {ExpIntegralEi}(\log (x)-2)-\operatorname {ExpIntegralEi}(2 \log (x))-30 \operatorname {LogIntegral}(x)+2 x-\frac {50 \log ^2(\log (x)-2)}{\log ^2(x)}-10 \log ^2(\log (x)-2)+\log ^2(x)+\frac {10 x}{\log ^2(x)}+\frac {25}{\log ^2(x)}+\frac {25 \log ^4(\log (x)-2)}{\log ^2(x)}+\frac {25}{2} \log (2-\log (x))-\frac {25}{2} \log (\log (x))+\frac {20 x}{\log (x)}+\frac {25}{\log (x)}\)

Input:

Int[(100 + 40*x + 4*x^2 + (-50 - 40*x - 6*x^2)*Log[x] + (10*x + 2*x^2)*Log 
[x]^2 - 4*x*Log[x]^3 + (-4 + 2*x)*Log[x]^4 + 2*Log[x]^5 + ((-100 - 20*x)*L 
og[x] - 20*Log[x]^3)*Log[-2 + Log[x]] + (-200 - 40*x + (100 + 40*x)*Log[x] 
 - 10*x*Log[x]^2)*Log[-2 + Log[x]]^2 + 100*Log[x]*Log[-2 + Log[x]]^3 + (10 
0 - 50*Log[x])*Log[-2 + Log[x]]^4)/(-2*x*Log[x]^3 + x*Log[x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03

method	result	size
risch	\(\frac {25 \ln \left (\ln \left (x \right )-2\right )^{4}}{\ln \left (x \right )^{2}}-\frac {10 \left (\ln \left (x \right )^{2}+x +5\right ) \ln \left (\ln \left (x \right )-2\right )^{2}}{\ln \left (x \right )^{2}}+\frac {\ln \left (x \right )^{4}+2 x \ln \left (x \right )^{2}+x^{2}+10 x +25}{\ln \left (x \right )^{2}}\)	\(59\)
parallelrisch	\(-\frac {-4 \ln \left (x \right )^{4}+40 \ln \left (x \right )^{2} \ln \left (\ln \left (x \right )-2\right )^{2}-100 \ln \left (\ln \left (x \right )-2\right )^{4}-8 x \ln \left (x \right )^{2}+40 x \ln \left (\ln \left (x \right )-2\right )^{2}-100-4 x^{2}-16 \ln \left (x \right )^{2}+200 \ln \left (\ln \left (x \right )-2\right )^{2}-40 x}{4 \ln \left (x \right )^{2}}\)	\(77\)

Input:

int(((-50*ln(x)+100)*ln(ln(x)-2)^4+100*ln(x)*ln(ln(x)-2)^3+(-10*x*ln(x)^2+ 
(40*x+100)*ln(x)-40*x-200)*ln(ln(x)-2)^2+(-20*ln(x)^3+(-20*x-100)*ln(x))*l 
n(ln(x)-2)+2*ln(x)^5+(2*x-4)*ln(x)^4-4*x*ln(x)^3+(2*x^2+10*x)*ln(x)^2+(-6* 
x^2-40*x-50)*ln(x)+4*x^2+40*x+100)/(x*ln(x)^4-2*x*ln(x)^3),x,method=_RETUR 
NVERBOSE)
 

Output:

25/ln(x)^2*ln(ln(x)-2)^4-10*(ln(x)^2+x+5)/ln(x)^2*ln(ln(x)-2)^2+(ln(x)^4+2 
*x*ln(x)^2+x^2+10*x+25)/ln(x)^2
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\frac {\log \left (x\right )^{4} + 25 \, \log \left (\log \left (x\right ) - 2\right )^{4} + 2 \, x \log \left (x\right )^{2} - 10 \, {\left (\log \left (x\right )^{2} + x + 5\right )} \log \left (\log \left (x\right ) - 2\right )^{2} + x^{2} + 10 \, x + 25}{\log \left (x\right )^{2}} \] Input:

integrate(((-50*log(x)+100)*log(log(x)-2)^4+100*log(x)*log(log(x)-2)^3+(-1 
0*x*log(x)^2+(40*x+100)*log(x)-40*x-200)*log(log(x)-2)^2+(-20*log(x)^3+(-2 
0*x-100)*log(x))*log(log(x)-2)+2*log(x)^5+(2*x-4)*log(x)^4-4*x*log(x)^3+(2 
*x^2+10*x)*log(x)^2+(-6*x^2-40*x-50)*log(x)+4*x^2+40*x+100)/(x*log(x)^4-2* 
x*log(x)^3),x, algorithm="fricas")
 

Output:

(log(x)^4 + 25*log(log(x) - 2)^4 + 2*x*log(x)^2 - 10*(log(x)^2 + x + 5)*lo 
g(log(x) - 2)^2 + x^2 + 10*x + 25)/log(x)^2
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((-50*ln(x)+100)*ln(ln(x)-2)**4+100*ln(x)*ln(ln(x)-2)**3+(-10*x* 
ln(x)**2+(40*x+100)*ln(x)-40*x-200)*ln(ln(x)-2)**2+(-20*ln(x)**3+(-20*x-10 
0)*ln(x))*ln(ln(x)-2)+2*ln(x)**5+(2*x-4)*ln(x)**4-4*x*ln(x)**3+(2*x**2+10* 
x)*ln(x)**2+(-6*x**2-40*x-50)*ln(x)+4*x**2+40*x+100)/(x*ln(x)**4-2*x*ln(x) 
**3),x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (30) = 60\).

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\frac {\log \left (x\right )^{4} + 25 \, \log \left (\log \left (x\right ) - 2\right )^{4} + 2 \, x \log \left (x\right )^{2} - 10 \, {\left (\log \left (x\right )^{2} + x + 5\right )} \log \left (\log \left (x\right ) - 2\right )^{2} + x^{2} + 10 \, x - 25 \, \log \left (x\right )}{\log \left (x\right )^{2}} + \frac {25 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )^{2}} \] Input:

integrate(((-50*log(x)+100)*log(log(x)-2)^4+100*log(x)*log(log(x)-2)^3+(-1 
0*x*log(x)^2+(40*x+100)*log(x)-40*x-200)*log(log(x)-2)^2+(-20*log(x)^3+(-2 
0*x-100)*log(x))*log(log(x)-2)+2*log(x)^5+(2*x-4)*log(x)^4-4*x*log(x)^3+(2 
*x^2+10*x)*log(x)^2+(-6*x^2-40*x-50)*log(x)+4*x^2+40*x+100)/(x*log(x)^4-2* 
x*log(x)^3),x, algorithm="maxima")
 

Output:

(log(x)^4 + 25*log(log(x) - 2)^4 + 2*x*log(x)^2 - 10*(log(x)^2 + x + 5)*lo 
g(log(x) - 2)^2 + x^2 + 10*x - 25*log(x))/log(x)^2 + 25*(log(x) + 1)/log(x 
)^2
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=-10 \, {\left (\frac {x + 5}{\log \left (x\right )^{2}} + 1\right )} \log \left (\log \left (x\right ) - 2\right )^{2} + \log \left (x\right )^{2} + \frac {25 \, \log \left (\log \left (x\right ) - 2\right )^{4}}{\log \left (x\right )^{2}} + 2 \, x + \frac {x^{2} + 10 \, x + 25}{\log \left (x\right )^{2}} \] Input:

integrate(((-50*log(x)+100)*log(log(x)-2)^4+100*log(x)*log(log(x)-2)^3+(-1 
0*x*log(x)^2+(40*x+100)*log(x)-40*x-200)*log(log(x)-2)^2+(-20*log(x)^3+(-2 
0*x-100)*log(x))*log(log(x)-2)+2*log(x)^5+(2*x-4)*log(x)^4-4*x*log(x)^3+(2 
*x^2+10*x)*log(x)^2+(-6*x^2-40*x-50)*log(x)+4*x^2+40*x+100)/(x*log(x)^4-2* 
x*log(x)^3),x, algorithm="giac")
 

Output:

-10*((x + 5)/log(x)^2 + 1)*log(log(x) - 2)^2 + log(x)^2 + 25*log(log(x) - 
2)^4/log(x)^2 + 2*x + (x^2 + 10*x + 25)/log(x)^2
 

Mupad [B] (verification not implemented)

Time = 1.99 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.00 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=7\,x+{\ln \left (x\right )}^2+\frac {x\,\left (x+5\right )-x\,\ln \left (x\right )\,\left (2\,x+5\right )}{\ln \left (x\right )}-{\ln \left (\ln \left (x\right )-2\right )}^2\,\left (\frac {10\,x+50}{{\ln \left (x\right )}^2}+10\right )+\frac {25\,{\ln \left (\ln \left (x\right )-2\right )}^4}{{\ln \left (x\right )}^2}+\frac {{\left (x+5\right )}^2-x\,\ln \left (x\right )\,\left (x+5\right )}{{\ln \left (x\right )}^2}+2\,x^2 \] Input:

int(-(40*x + log(x)^2*(10*x + 2*x^2) - log(log(x) - 2)^4*(50*log(x) - 100) 
 - 4*x*log(x)^3 - log(log(x) - 2)^2*(40*x + 10*x*log(x)^2 - log(x)*(40*x + 
 100) + 200) + 2*log(x)^5 + 100*log(log(x) - 2)^3*log(x) - log(log(x) - 2) 
*(20*log(x)^3 + log(x)*(20*x + 100)) - log(x)*(40*x + 6*x^2 + 50) + 4*x^2 
+ log(x)^4*(2*x - 4) + 100)/(2*x*log(x)^3 - x*log(x)^4),x)
 

Output:

7*x + log(x)^2 + (x*(x + 5) - x*log(x)*(2*x + 5))/log(x) - log(log(x) - 2) 
^2*((10*x + 50)/log(x)^2 + 10) + (25*log(log(x) - 2)^4)/log(x)^2 + ((x + 5 
)^2 - x*log(x)*(x + 5))/log(x)^2 + 2*x^2
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\frac {25 \mathrm {log}\left (\mathrm {log}\left (x \right )-2\right )^{4}-10 \mathrm {log}\left (\mathrm {log}\left (x \right )-2\right )^{2} \mathrm {log}\left (x \right )^{2}-10 \mathrm {log}\left (\mathrm {log}\left (x \right )-2\right )^{2} x -50 \mathrm {log}\left (\mathrm {log}\left (x \right )-2\right )^{2}+\mathrm {log}\left (x \right )^{4}+2 \mathrm {log}\left (x \right )^{2} x +x^{2}+10 x +25}{\mathrm {log}\left (x \right )^{2}} \] Input:

int(((-50*log(x)+100)*log(log(x)-2)^4+100*log(x)*log(log(x)-2)^3+(-10*x*lo 
g(x)^2+(40*x+100)*log(x)-40*x-200)*log(log(x)-2)^2+(-20*log(x)^3+(-20*x-10 
0)*log(x))*log(log(x)-2)+2*log(x)^5+(2*x-4)*log(x)^4-4*x*log(x)^3+(2*x^2+1 
0*x)*log(x)^2+(-6*x^2-40*x-50)*log(x)+4*x^2+40*x+100)/(x*log(x)^4-2*x*log( 
x)^3),x)
 

Output:

(25*log(log(x) - 2)**4 - 10*log(log(x) - 2)**2*log(x)**2 - 10*log(log(x) - 
 2)**2*x - 50*log(log(x) - 2)**2 + log(x)**4 + 2*log(x)**2*x + x**2 + 10*x 
 + 25)/log(x)**2