\(\int \frac {-1-x+e^{5 x} (1+x)+(2-2 e^{5 x}) \log (x)+(-x+e^{5 x} x+(1-e^{5 x}) \log (x)) \log (-x+\log (x))+(2 x+e^{5 x} (-2 x-10 x^2)+(-2+e^{5 x} (2+10 x)) \log (x)+(x+e^{5 x} (-x-5 x^2)+(-1+e^{5 x} (1+5 x)) \log (x)) \log (-x+\log (x))) \log (-\frac {x}{2+\log (-x+\log (x))})}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2(-\frac {x}{2+\log (-x+\log (x))})} \, dx\) [1552]

Optimal result

Integrand size = 191, antiderivative size = 29 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx=1+\frac {\left (-1+e^{5 x}\right ) x}{\log \left (\frac {x}{-2-\log (-x+\log (x))}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx=\frac {\left (-1+e^{5 x}\right ) x}{\log \left (-\frac {x}{2+\log (-x+\log (x))}\right )} \] Input:

Integrate[(-1 - x + E^(5*x)*(1 + x) + (2 - 2*E^(5*x))*Log[x] + (-x + E^(5* 
x)*x + (1 - E^(5*x))*Log[x])*Log[-x + Log[x]] + (2*x + E^(5*x)*(-2*x - 10* 
x^2) + (-2 + E^(5*x)*(2 + 10*x))*Log[x] + (x + E^(5*x)*(-x - 5*x^2) + (-1 
+ E^(5*x)*(1 + 5*x))*Log[x])*Log[-x + Log[x]])*Log[-(x/(2 + Log[-x + Log[x 
]]))])/((-2*x + 2*Log[x] + (-x + Log[x])*Log[-x + Log[x]])*Log[-(x/(2 + Lo 
g[-x + Log[x]]))]^2),x]
 

Output:

((-1 + E^(5*x))*x)/Log[-(x/(2 + Log[-x + Log[x]]))]
 

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.

\(\displaystyle \int \frac {\left (e^{5 x} \left (-10 x^2-2 x\right )+\left (e^{5 x} \left (-5 x^2-x\right )+x+\left (e^{5 x} (5 x+1)-1\right ) \log (x)\right ) \log (\log (x)-x)+2 x+\left (e^{5 x} (10 x+2)-2\right ) \log (x)\right ) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )-x+e^{5 x} (x+1)+\left (2-2 e^{5 x}\right ) \log (x)+\left (e^{5 x} x-x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (\log (x)-x)-1}{(-2 x+2 \log (x)+(\log (x)-x) \log (\log (x)-x)) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-\left (e^{5 x} \left (-10 x^2-2 x\right )+\left (e^{5 x} \left (-5 x^2-x\right )+x+\left (e^{5 x} (5 x+1)-1\right ) \log (x)\right ) \log (\log (x)-x)+2 x+\left (e^{5 x} (10 x+2)-2\right ) \log (x)\right ) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )+x-e^{5 x} (x+1)-\left (2-2 e^{5 x}\right ) \log (x)-\left (e^{5 x} x-x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (\log (x)-x)+1}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x-e^{5 x} (x+1)+2 \left (e^{5 x}-1\right ) \log (x)-\left (e^{5 x}-1\right ) (x-\log (x)) \log (\log (x)-x)+\left (e^{5 x} (5 x+1)-1\right ) (x-\log (x)) (\log (\log (x)-x)+2) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )+1}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{5 x} \left (5 x^2 \log (\log (x)-x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )+10 x^2 \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )-x-x \log (\log (x)-x)-10 x \log (x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )-5 x \log (x) \log (\log (x)-x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )+x \log (\log (x)-x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )+2 x \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )+2 \log (x)+\log (x) \log (\log (x)-x)-2 \log (x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )-\log (x) \log (\log (x)-x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )-1\right )}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}+\frac {x}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}-\frac {2 \log (x)}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}+\frac {\log (\log (x)-x)}{(\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}+\frac {1}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}-\frac {1}{\log \left (-\frac {x}{\log (\log (x)-x)+2}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}dx+\int \frac {x}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}dx-2 \int \frac {\log (x)}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}dx+\int \frac {\log (\log (x)-x)}{(\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}dx-\int \frac {1}{\log \left (-\frac {x}{\log (\log (x)-x)+2}\right )}dx+\frac {e^{5 x} \left (x^2 \log (\log (x)-x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )+2 x^2 \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )-2 x \log (x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )-x \log (x) \log (\log (x)-x) \log \left (-\frac {x}{\log (\log (x)-x)+2}\right )\right )}{(x-\log (x)) (\log (\log (x)-x)+2) \log ^2\left (-\frac {x}{\log (\log (x)-x)+2}\right )}\)

Input:

Int[(-1 - x + E^(5*x)*(1 + x) + (2 - 2*E^(5*x))*Log[x] + (-x + E^(5*x)*x + 
 (1 - E^(5*x))*Log[x])*Log[-x + Log[x]] + (2*x + E^(5*x)*(-2*x - 10*x^2) + 
 (-2 + E^(5*x)*(2 + 10*x))*Log[x] + (x + E^(5*x)*(-x - 5*x^2) + (-1 + E^(5 
*x)*(1 + 5*x))*Log[x])*Log[-x + Log[x]])*Log[-(x/(2 + Log[-x + Log[x]]))]) 
/((-2*x + 2*Log[x] + (-x + Log[x])*Log[-x + Log[x]])*Log[-(x/(2 + Log[-x + 
 Log[x]]))]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 49.97 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(-\frac {-24 x \,{\mathrm e}^{5 x}+24 x}{24 \ln \left (-\frac {x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )}\)	\(31\)
risch	\(\frac {2 i x \left ({\mathrm e}^{5 x}-1\right )}{\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )-x \right )+2}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )-x \right )+2}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{3}+2 \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2}-2 \pi -2 i \ln \left (\ln \left (\ln \left (x \right )-x \right )+2\right )+2 i \ln \left (x \right )}\)	\(177\)

Input:

int((((((1+5*x)*exp(5*x)-1)*ln(x)+(-5*x^2-x)*exp(5*x)+x)*ln(ln(x)-x)+((10* 
x+2)*exp(5*x)-2)*ln(x)+(-10*x^2-2*x)*exp(5*x)+2*x)*ln(-x/(ln(ln(x)-x)+2))+ 
((-exp(5*x)+1)*ln(x)+x*exp(5*x)-x)*ln(ln(x)-x)+(-2*exp(5*x)+2)*ln(x)+(1+x) 
*exp(5*x)-x-1)/((ln(x)-x)*ln(ln(x)-x)+2*ln(x)-2*x)/ln(-x/(ln(ln(x)-x)+2))^ 
2,x,method=_RETURNVERBOSE)
 

Output:

-1/24*(-24*x*exp(5*x)+24*x)/ln(-x/(ln(ln(x)-x)+2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx=\frac {x e^{\left (5 \, x\right )} - x}{\log \left (-\frac {x}{\log \left (-x + \log \left (x\right )\right ) + 2}\right )} \] Input:

integrate((((((1+5*x)*exp(5*x)-1)*log(x)+(-5*x^2-x)*exp(5*x)+x)*log(log(x) 
-x)+((10*x+2)*exp(5*x)-2)*log(x)+(-10*x^2-2*x)*exp(5*x)+2*x)*log(-x/(log(l 
og(x)-x)+2))+((-exp(5*x)+1)*log(x)+x*exp(5*x)-x)*log(log(x)-x)+(-2*exp(5*x 
)+2)*log(x)+(1+x)*exp(5*x)-x-1)/((log(x)-x)*log(log(x)-x)+2*log(x)-2*x)/lo 
g(-x/(log(log(x)-x)+2))^2,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 4.74 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx=\frac {x e^{5 x}}{\log {\left (- \frac {x}{\log {\left (- x + \log {\left (x \right )} \right )} + 2} \right )}} - \frac {x}{\log {\left (- \frac {x}{\log {\left (- x + \log {\left (x \right )} \right )} + 2} \right )}} \] Input:

integrate((((((1+5*x)*exp(5*x)-1)*ln(x)+(-5*x**2-x)*exp(5*x)+x)*ln(ln(x)-x 
)+((10*x+2)*exp(5*x)-2)*ln(x)+(-10*x**2-2*x)*exp(5*x)+2*x)*ln(-x/(ln(ln(x) 
-x)+2))+((-exp(5*x)+1)*ln(x)+x*exp(5*x)-x)*ln(ln(x)-x)+(-2*exp(5*x)+2)*ln( 
x)+(1+x)*exp(5*x)-x-1)/((ln(x)-x)*ln(ln(x)-x)+2*ln(x)-2*x)/ln(-x/(ln(ln(x) 
-x)+2))**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx=\frac {x e^{\left (5 \, x\right )} - x}{\log \left (x\right ) - \log \left (-\log \left (-x + \log \left (x\right )\right ) - 2\right )} \] Input:

integrate((((((1+5*x)*exp(5*x)-1)*log(x)+(-5*x^2-x)*exp(5*x)+x)*log(log(x) 
-x)+((10*x+2)*exp(5*x)-2)*log(x)+(-10*x^2-2*x)*exp(5*x)+2*x)*log(-x/(log(l 
og(x)-x)+2))+((-exp(5*x)+1)*log(x)+x*exp(5*x)-x)*log(log(x)-x)+(-2*exp(5*x 
)+2)*log(x)+(1+x)*exp(5*x)-x-1)/((log(x)-x)*log(log(x)-x)+2*log(x)-2*x)/lo 
g(-x/(log(log(x)-x)+2))^2,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1672 vs. \(2 (27) = 54\).

Time = 1.19 (sec) , antiderivative size = 1672, normalized size of antiderivative = 57.66 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx=\text {Too large to display} \] Input:

integrate((((((1+5*x)*exp(5*x)-1)*log(x)+(-5*x^2-x)*exp(5*x)+x)*log(log(x) 
-x)+((10*x+2)*exp(5*x)-2)*log(x)+(-10*x^2-2*x)*exp(5*x)+2*x)*log(-x/(log(l 
og(x)-x)+2))+((-exp(5*x)+1)*log(x)+x*exp(5*x)-x)*log(log(x)-x)+(-2*exp(5*x 
)+2)*log(x)+(1+x)*exp(5*x)-x-1)/((log(x)-x)*log(log(x)-x)+2*log(x)-2*x)/lo 
g(-x/(log(log(x)-x)+2))^2,x, algorithm="giac")
 

Output:

-2*(x*e^(5*x)*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/(x - log(abs(x))))*sgn( 
-pi + pi*sgn(x))*sgn(x - log(abs(x))) - pi*arctan(-1/2*(pi - pi*sgn(x))/(x 
 - log(abs(x))))*sgn(-pi + pi*sgn(x)) + 1/2*pi^2*sgn(x - log(abs(x))) + 1/ 
2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/(x - log(abs(x))))^2 + 1/4*log(-1/2* 
pi^2*sgn(x) + 1/2*pi^2 + x^2 - 2*x*log(abs(x)) + log(abs(x))^2)^2 + 2*log( 
-1/2*pi^2*sgn(x) + 1/2*pi^2 + x^2 - 2*x*log(abs(x)) + log(abs(x))^2) + 4) 
- 2*x*e^(5*x)*log(abs(x)) - x*log(-pi*arctan(-1/2*(pi - pi*sgn(x))/(x - lo 
g(abs(x))))*sgn(-pi + pi*sgn(x))*sgn(x - log(abs(x))) - pi*arctan(-1/2*(pi 
 - pi*sgn(x))/(x - log(abs(x))))*sgn(-pi + pi*sgn(x)) + 1/2*pi^2*sgn(x - l 
og(abs(x))) + 1/2*pi^2 + arctan(-1/2*(pi - pi*sgn(x))/(x - log(abs(x))))^2 
 + 1/4*log(-1/2*pi^2*sgn(x) + 1/2*pi^2 + x^2 - 2*x*log(abs(x)) + log(abs(x 
))^2)^2 + 2*log(-1/2*pi^2*sgn(x) + 1/2*pi^2 + x^2 - 2*x*log(abs(x)) + log( 
abs(x))^2) + 4) + 2*x*log(abs(x)))/(2*pi^2*sgn(pi*sgn(-pi + pi*sgn(x))*sgn 
(x - log(abs(x))) + pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x 
))/(x - log(abs(x)))))*sgn(x)*sgn(log(-1/2*pi^2*sgn(x) + 1/2*pi^2 + x^2 - 
2*x*log(abs(x)) + log(abs(x))^2) + 4) - 2*pi^2*sgn(pi*sgn(-pi + pi*sgn(x)) 
*sgn(x - log(abs(x))) + pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*s 
gn(x))/(x - log(abs(x)))))*sgn(x) - 6*pi^2*sgn(pi*sgn(-pi + pi*sgn(x))*sgn 
(x - log(abs(x))) + pi*sgn(-pi + pi*sgn(x)) - 2*arctan(-1/2*(pi - pi*sgn(x 
))/(x - log(abs(x)))))*sgn(log(-1/2*pi^2*sgn(x) + 1/2*pi^2 + x^2 - 2*x*...
 

Mupad [B] (verification not implemented)

Time = 3.33 (sec) , antiderivative size = 266, normalized size of antiderivative = 9.17 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx={\mathrm {e}}^{5\,x}\,\left (5\,x^2+x\right )-x+\frac {x\,\left ({\mathrm {e}}^{5\,x}-1\right )-\frac {x\,\ln \left (-\frac {x}{\ln \left (\ln \left (x\right )-x\right )+2}\right )\,\left (x-\ln \left (x\right )\right )\,\left (\ln \left (\ln \left (x\right )-x\right )+2\right )\,\left ({\mathrm {e}}^{5\,x}+5\,x\,{\mathrm {e}}^{5\,x}-1\right )}{x-2\,\ln \left (x\right )+x\,\ln \left (\ln \left (x\right )-x\right )-\ln \left (\ln \left (x\right )-x\right )\,\ln \left (x\right )+1}}{\ln \left (-\frac {x}{\ln \left (\ln \left (x\right )-x\right )+2}\right )}-\frac {x^3\,\ln \left (x\right )-x^2\,\ln \left (x\right )+x^2\,{\mathrm {e}}^{5\,x}+x^3\,{\mathrm {e}}^{5\,x}-16\,x^4\,{\mathrm {e}}^{5\,x}+19\,x^5\,{\mathrm {e}}^{5\,x}-5\,x^6\,{\mathrm {e}}^{5\,x}-x^2+4\,x^3-4\,x^4+x^5+x^2\,{\mathrm {e}}^{5\,x}\,\ln \left (x\right )+4\,x^3\,{\mathrm {e}}^{5\,x}\,\ln \left (x\right )-5\,x^4\,{\mathrm {e}}^{5\,x}\,\ln \left (x\right )}{\left (x-2\,\ln \left (x\right )+\ln \left (\ln \left (x\right )-x\right )\,\left (x-\ln \left (x\right )\right )+1\right )\,\left (x+x\,\ln \left (x\right )-3\,x^2+x^3\right )} \] Input:

int((x + log(x)*(2*exp(5*x) - 2) - exp(5*x)*(x + 1) - log(-x/(log(log(x) - 
 x) + 2))*(2*x - exp(5*x)*(2*x + 10*x^2) + log(x)*(exp(5*x)*(10*x + 2) - 2 
) + log(log(x) - x)*(x + log(x)*(exp(5*x)*(5*x + 1) - 1) - exp(5*x)*(x + 5 
*x^2))) + log(log(x) - x)*(x - x*exp(5*x) + log(x)*(exp(5*x) - 1)) + 1)/(l 
og(-x/(log(log(x) - x) + 2))^2*(2*x - 2*log(x) + log(log(x) - x)*(x - log( 
x)))),x)
 

Output:

exp(5*x)*(x + 5*x^2) - x + (x*(exp(5*x) - 1) - (x*log(-x/(log(log(x) - x) 
+ 2))*(x - log(x))*(log(log(x) - x) + 2)*(exp(5*x) + 5*x*exp(5*x) - 1))/(x 
 - 2*log(x) + x*log(log(x) - x) - log(log(x) - x)*log(x) + 1))/log(-x/(log 
(log(x) - x) + 2)) - (x^3*log(x) - x^2*log(x) + x^2*exp(5*x) + x^3*exp(5*x 
) - 16*x^4*exp(5*x) + 19*x^5*exp(5*x) - 5*x^6*exp(5*x) - x^2 + 4*x^3 - 4*x 
^4 + x^5 + x^2*exp(5*x)*log(x) + 4*x^3*exp(5*x)*log(x) - 5*x^4*exp(5*x)*lo 
g(x))/((x - 2*log(x) + log(log(x) - x)*(x - log(x)) + 1)*(x + x*log(x) - 3 
*x^2 + x^3))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx=\frac {x \left (e^{5 x}-1\right )}{\mathrm {log}\left (-\frac {x}{\mathrm {log}\left (\mathrm {log}\left (x \right )-x \right )+2}\right )} \] Input:

int((((((1+5*x)*exp(5*x)-1)*log(x)+(-5*x^2-x)*exp(5*x)+x)*log(log(x)-x)+(( 
10*x+2)*exp(5*x)-2)*log(x)+(-10*x^2-2*x)*exp(5*x)+2*x)*log(-x/(log(log(x)- 
x)+2))+((-exp(5*x)+1)*log(x)+x*exp(5*x)-x)*log(log(x)-x)+(-2*exp(5*x)+2)*l 
og(x)+(1+x)*exp(5*x)-x-1)/((log(x)-x)*log(log(x)-x)+2*log(x)-2*x)/log(-x/( 
log(log(x)-x)+2))^2,x)
 

Output:

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