\(\int \frac {-625 x-1250 x^3+1875 x^4+e^x (625+1250 x^2-1875 x^3)+(-2+2 e^x) \log (e^x-x)+(1250 x^2-2500 x^3+e^x (-1250 x+2500 x^2)) \log (x)+(-625 e^x x+625 x^2) \log ^2(x)+\log ^2(e^x-x) (-2 x^2+3 x^3+e^x (2 x-3 x^2)+(2 x-4 x^2+e^x (-2+4 x)) \log (x)+(-e^x+x) \log ^2(x))}{625 e^x x-625 x^2+(e^x-x) \log ^2(e^x-x)} \, dx\) [1789]

Optimal result

Integrand size = 191, antiderivative size = 31 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x (-x+\log (x))^2+\log \left (-x-\frac {1}{625} \log ^2\left (e^x-x\right )\right ) \] Output:

ln(-1/625*ln(exp(x)-x)^2-x)-x*(ln(x)-x)^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^3+2 x^2 \log (x)-x \log ^2(x)+\log \left (625 x+\log ^2\left (e^x-x\right )\right ) \] Input:

Integrate[(-625*x - 1250*x^3 + 1875*x^4 + E^x*(625 + 1250*x^2 - 1875*x^3) 
+ (-2 + 2*E^x)*Log[E^x - x] + (1250*x^2 - 2500*x^3 + E^x*(-1250*x + 2500*x 
^2))*Log[x] + (-625*E^x*x + 625*x^2)*Log[x]^2 + Log[E^x - x]^2*(-2*x^2 + 3 
*x^3 + E^x*(2*x - 3*x^2) + (2*x - 4*x^2 + E^x*(-2 + 4*x))*Log[x] + (-E^x + 
 x)*Log[x]^2))/(625*E^x*x - 625*x^2 + (E^x - x)*Log[E^x - x]^2),x]
 

Output:

-x^3 + 2*x^2*Log[x] - x*Log[x]^2 + Log[625*x + Log[E^x - 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.

\(\displaystyle \int \frac {1875 x^4-1250 x^3+\left (625 x^2-625 e^x x\right ) \log ^2(x)+e^x \left (-1875 x^3+1250 x^2+625\right )+\log ^2\left (e^x-x\right ) \left (3 x^3-2 x^2+e^x \left (2 x-3 x^2\right )+\left (-4 x^2+2 x+e^x (4 x-2)\right ) \log (x)+\left (x-e^x\right ) \log ^2(x)\right )+\left (-2500 x^3+1250 x^2+e^x \left (2500 x^2-1250 x\right )\right ) \log (x)-625 x+\left (2 e^x-2\right ) \log \left (e^x-x\right )}{-625 x^2+625 e^x x+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {1875 x^4-1250 x^3+\left (625 x^2-625 e^x x\right ) \log ^2(x)+e^x \left (-1875 x^3+1250 x^2+625\right )+\log ^2\left (e^x-x\right ) \left (3 x^3-2 x^2+e^x \left (2 x-3 x^2\right )+\left (-4 x^2+2 x+e^x (4 x-2)\right ) \log (x)+\left (x-e^x\right ) \log ^2(x)\right )+\left (-2500 x^3+1250 x^2+e^x \left (2500 x^2-1250 x\right )\right ) \log (x)-625 x+\left (2 e^x-2\right ) \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-625 \left (e^x-x\right ) \left (3 x^3-2 x^2+x \log ^2(x)+2 (1-2 x) x \log (x)-1\right )-\left (\left (e^x-x\right ) \left (x (3 x-2)+\log ^2(x)+(2-4 x) \log (x)\right ) \log ^2\left (e^x-x\right )\right )+2 \left (e^x-1\right ) \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-1875 x^3+1250 x^2-3 x^2 \log ^2\left (e^x-x\right )+2500 x^2 \log (x)+2 x \log ^2\left (e^x-x\right )-625 x \log ^2(x)+4 x \log ^2\left (e^x-x\right ) \log (x)-\log ^2\left (e^x-x\right ) \log ^2(x)-2 \log ^2\left (e^x-x\right ) \log (x)-1250 x \log (x)+2 \log \left (e^x-x\right )+625}{625 x+\log ^2\left (e^x-x\right )}+\frac {2 (x-1) \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 625 \int \frac {1}{\log ^2\left (e^x-x\right )+625 x}dx+2 \int \frac {\log \left (e^x-x\right )}{\log ^2\left (e^x-x\right )+625 x}dx-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (\log ^2\left (e^x-x\right )+625 x\right )}dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (\log ^2\left (e^x-x\right )+625 x\right )}dx-x^3-x \log ^2(x)+2 x \log (x)+\frac {1}{2} (1-2 x)^2 \log (x)-\frac {\log (x)}{2}\)

Input:

Int[(-625*x - 1250*x^3 + 1875*x^4 + E^x*(625 + 1250*x^2 - 1875*x^3) + (-2 
+ 2*E^x)*Log[E^x - x] + (1250*x^2 - 2500*x^3 + E^x*(-1250*x + 2500*x^2))*L 
og[x] + (-625*E^x*x + 625*x^2)*Log[x]^2 + Log[E^x - x]^2*(-2*x^2 + 3*x^3 + 
 E^x*(2*x - 3*x^2) + (2*x - 4*x^2 + E^x*(-2 + 4*x))*Log[x] + (-E^x + x)*Lo 
g[x]^2))/(625*E^x*x - 625*x^2 + (E^x - x)*Log[E^x - x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 96.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13

method	result	size
risch	\(-x^{3}+2 x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+\ln \left (\ln \left ({\mathrm e}^{x}-x \right )^{2}+625 x \right )\)	\(35\)
parallelrisch	\(-x^{3}+2 x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+\ln \left (\frac {\ln \left ({\mathrm e}^{x}-x \right )^{2}}{625}+x \right )\)	\(35\)

Input:

int((((x-exp(x))*ln(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*ln(x)+(-3*x^2+2*x)*exp 
(x)+3*x^3-2*x^2)*ln(exp(x)-x)^2+(2*exp(x)-2)*ln(exp(x)-x)+(-625*exp(x)*x+6 
25*x^2)*ln(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*ln(x)+(-1875* 
x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)*ln(exp(x)-x) 
^2+625*exp(x)*x-625*x^2),x,method=_RETURNVERBOSE)
 

Output:

-x^3+2*x^2*ln(x)-x*ln(x)^2+ln(ln(exp(x)-x)^2+625*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^{3} + 2 \, x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \] Input:

integrate((((x-exp(x))*log(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*log(x)+(-3*x^2+ 
2*x)*exp(x)+3*x^3-2*x^2)*log(exp(x)-x)^2+(2*exp(x)-2)*log(exp(x)-x)+(-625* 
exp(x)*x+625*x^2)*log(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*lo 
g(x)+(-1875*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)* 
log(exp(x)-x)^2+625*exp(x)*x-625*x^2),x, algorithm="fricas")
 

Output:

-x^3 + 2*x^2*log(x) - x*log(x)^2 + log(log(-x + e^x)^2 + 625*x)
 

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=- x^{3} + 2 x^{2} \log {\left (x \right )} - x \log {\left (x \right )}^{2} + \log {\left (625 x + \log {\left (- x + e^{x} \right )}^{2} \right )} \] Input:

integrate((((x-exp(x))*ln(x)**2+((4*x-2)*exp(x)-4*x**2+2*x)*ln(x)+(-3*x**2 
+2*x)*exp(x)+3*x**3-2*x**2)*ln(exp(x)-x)**2+(2*exp(x)-2)*ln(exp(x)-x)+(-62 
5*exp(x)*x+625*x**2)*ln(x)**2+((2500*x**2-1250*x)*exp(x)-2500*x**3+1250*x* 
*2)*ln(x)+(-1875*x**3+1250*x**2+625)*exp(x)+1875*x**4-1250*x**3-625*x)/((e 
xp(x)-x)*ln(exp(x)-x)**2+625*exp(x)*x-625*x**2),x)
 

Output:

-x**3 + 2*x**2*log(x) - x*log(x)**2 + log(625*x + log(-x + exp(x))**2)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^{3} + 2 \, x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \] Input:

integrate((((x-exp(x))*log(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*log(x)+(-3*x^2+ 
2*x)*exp(x)+3*x^3-2*x^2)*log(exp(x)-x)^2+(2*exp(x)-2)*log(exp(x)-x)+(-625* 
exp(x)*x+625*x^2)*log(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*lo 
g(x)+(-1875*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)* 
log(exp(x)-x)^2+625*exp(x)*x-625*x^2),x, algorithm="maxima")
 

Output:

-x^3 + 2*x^2*log(x) - x*log(x)^2 + log(log(-x + e^x)^2 + 625*x)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^{3} + 2 \, x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \] Input:

integrate((((x-exp(x))*log(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*log(x)+(-3*x^2+ 
2*x)*exp(x)+3*x^3-2*x^2)*log(exp(x)-x)^2+(2*exp(x)-2)*log(exp(x)-x)+(-625* 
exp(x)*x+625*x^2)*log(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*lo 
g(x)+(-1875*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)* 
log(exp(x)-x)^2+625*exp(x)*x-625*x^2),x, algorithm="giac")
 

Output:

-x^3 + 2*x^2*log(x) - x*log(x)^2 + log(log(-x + e^x)^2 + 625*x)
 

Mupad [B] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=\ln \left ({\ln \left ({\mathrm {e}}^x-x\right )}^2+625\,x\right )-x\,{\ln \left (x\right )}^2+2\,x^2\,\ln \left (x\right )-x^3 \] Input:

int((625*x - exp(x)*(1250*x^2 - 1875*x^3 + 625) - log(exp(x) - x)*(2*exp(x 
) - 2) - log(exp(x) - x)^2*(log(x)*(2*x + exp(x)*(4*x - 2) - 4*x^2) + exp( 
x)*(2*x - 3*x^2) + log(x)^2*(x - exp(x)) - 2*x^2 + 3*x^3) + log(x)*(exp(x) 
*(1250*x - 2500*x^2) - 1250*x^2 + 2500*x^3) + 1250*x^3 - 1875*x^4 + log(x) 
^2*(625*x*exp(x) - 625*x^2))/(log(exp(x) - x)^2*(x - exp(x)) - 625*x*exp(x 
) + 625*x^2),x)
 

Output:

log(625*x + log(exp(x) - x)^2) - x*log(x)^2 + 2*x^2*log(x) - x^3
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (e^{x}-x \right )^{2}+625 x \right )-\mathrm {log}\left (x \right )^{2} x +2 \,\mathrm {log}\left (x \right ) x^{2}-x^{3} \] Input:

int((((x-exp(x))*log(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*log(x)+(-3*x^2+2*x)*e 
xp(x)+3*x^3-2*x^2)*log(exp(x)-x)^2+(2*exp(x)-2)*log(exp(x)-x)+(-625*exp(x) 
*x+625*x^2)*log(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*log(x)+( 
-1875*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)*log(ex 
p(x)-x)^2+625*exp(x)*x-625*x^2),x)
 

Output:

log(log(e**x - x)**2 + 625*x) - log(x)**2*x + 2*log(x)*x**2 - x**3