\(\int \frac {e^4 (-1+17 x-4 x^2-12 x^3+3 x^4)+e^4 (4 x-3 x^3) \log (x)+(e^4 (20-5 x-12 x^2+3 x^3)+e^4 (5-3 x^2) \log (x)) \log (-4+x-\log (x))+(e^4 (-4 x+x^2)-e^4 x \log (x)+(e^4 (-4+x)-e^4 \log (x)) \log (-4+x-\log (x))) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx\) [1003]

Optimal result

Integrand size = 170, antiderivative size = 28 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=e^4 \left (9+x \left (5-x^2-\log (x+\log (-4+x-\log (x)))\right )\right ) \] Output:

((5-x^2-ln(ln(-ln(x)+x-4)+x))*x+9)*exp(4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=e^4 \left (5 x-x^3-x \log (x+\log (-4+x-\log (x)))\right ) \] Input:

Integrate[(E^4*(-1 + 17*x - 4*x^2 - 12*x^3 + 3*x^4) + E^4*(4*x - 3*x^3)*Lo 
g[x] + (E^4*(20 - 5*x - 12*x^2 + 3*x^3) + E^4*(5 - 3*x^2)*Log[x])*Log[-4 + 
 x - Log[x]] + (E^4*(-4*x + x^2) - E^4*x*Log[x] + (E^4*(-4 + x) - E^4*Log[ 
x])*Log[-4 + x - Log[x]])*Log[x + Log[-4 + x - Log[x]]])/(4*x - x^2 + x*Lo 
g[x] + (4 - x + Log[x])*Log[-4 + x - Log[x]]),x]
 

Output:

E^4*(5*x - x^3 - x*Log[x + Log[-4 + 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.

\(\displaystyle \int \frac {e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (x^2-4 x\right )-e^4 x \log (x)+\left (e^4 (x-4)-e^4 \log (x)\right ) \log (x-\log (x)-4)\right ) \log (x+\log (x-\log (x)-4))+\left (e^4 \left (5-3 x^2\right ) \log (x)+e^4 \left (3 x^3-12 x^2-5 x+20\right )\right ) \log (x-\log (x)-4)+e^4 \left (3 x^4-12 x^3-4 x^2+17 x-1\right )}{-x^2+4 x+x \log (x)+(-x+\log (x)+4) \log (x-\log (x)-4)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (x^2-4 x\right )-e^4 x \log (x)+\left (e^4 (x-4)-e^4 \log (x)\right ) \log (x-\log (x)-4)\right ) \log (x+\log (x-\log (x)-4))+\left (e^4 \left (5-3 x^2\right ) \log (x)+e^4 \left (3 x^3-12 x^2-5 x+20\right )\right ) \log (x-\log (x)-4)+e^4 \left (3 x^4-12 x^3-4 x^2+17 x-1\right )}{(-x+\log (x)+4) (x+\log (x-\log (x)-4))}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {e^4 \left (3 x^4-12 x^3-3 x^3 \log (x)+3 x^3 \log (x-\log (x)-4)-4 x^2-3 x^2 \log (x) \log (x-\log (x)-4)-12 x^2 \log (x-\log (x)-4)+17 x+4 x \log (x)-5 x \log (x-\log (x)-4)+5 \log (x) \log (x-\log (x)-4)+20 \log (x-\log (x)-4)-1\right )}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}-e^4 \log (x+\log (x-\log (x)-4))\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -e^4 \int \frac {x^2}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx+e^4 \int \frac {1}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx+3 e^4 \int \frac {x}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx+e^4 \int \frac {x \log (x)}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx-e^4 \int \log (x+\log (x-\log (x)-4))dx-e^4 x^3+5 e^4 x\)

Input:

Int[(E^4*(-1 + 17*x - 4*x^2 - 12*x^3 + 3*x^4) + E^4*(4*x - 3*x^3)*Log[x] + 
 (E^4*(20 - 5*x - 12*x^2 + 3*x^3) + E^4*(5 - 3*x^2)*Log[x])*Log[-4 + x - L 
og[x]] + (E^4*(-4*x + x^2) - E^4*x*Log[x] + (E^4*(-4 + x) - E^4*Log[x])*Lo 
g[-4 + x - Log[x]])*Log[x + Log[-4 + x - Log[x]]])/(4*x - x^2 + x*Log[x] + 
 (4 - x + Log[x])*Log[-4 + x - Log[x]]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 7.63 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
risch	\(-{\mathrm e}^{4} x \ln \left (\ln \left (-\ln \left (x \right )+x -4\right )+x \right )-{\mathrm e}^{4} x \left (x^{2}-5\right )\)	\(28\)
parallelrisch	\(-x^{3} {\mathrm e}^{4}-{\mathrm e}^{4} x \ln \left (\ln \left (-\ln \left (x \right )+x -4\right )+x \right )+5 x \,{\mathrm e}^{4}\)	\(30\)

Input:

int((((-exp(4)*ln(x)+(x-4)*exp(4))*ln(-ln(x)+x-4)-x*exp(4)*ln(x)+(x^2-4*x) 
*exp(4))*ln(ln(-ln(x)+x-4)+x)+((-3*x^2+5)*exp(4)*ln(x)+(3*x^3-12*x^2-5*x+2 
0)*exp(4))*ln(-ln(x)+x-4)+(-3*x^3+4*x)*exp(4)*ln(x)+(3*x^4-12*x^3-4*x^2+17 
*x-1)*exp(4))/((ln(x)-x+4)*ln(-ln(x)+x-4)+x*ln(x)-x^2+4*x),x,method=_RETUR 
NVERBOSE)
 

Output:

-exp(4)*x*ln(ln(-ln(x)+x-4)+x)-exp(4)*x*(x^2-5)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x e^{4} \log \left (x + \log \left (x - \log \left (x\right ) - 4\right )\right ) - {\left (x^{3} - 5 \, x\right )} e^{4} \] Input:

integrate((((-exp(4)*log(x)+(-4+x)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x 
)+(x^2-4*x)*exp(4))*log(log(-log(x)+x-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x 
^3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*x 
^4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^ 
2+4*x),x, algorithm="fricas")
 

Output:

-x*e^4*log(x + log(x - log(x) - 4)) - (x^3 - 5*x)*e^4
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((((-exp(4)*ln(x)+(-4+x)*exp(4))*ln(-ln(x)+x-4)-x*exp(4)*ln(x)+(x 
**2-4*x)*exp(4))*ln(ln(-ln(x)+x-4)+x)+((-3*x**2+5)*exp(4)*ln(x)+(3*x**3-12 
*x**2-5*x+20)*exp(4))*ln(-ln(x)+x-4)+(-3*x**3+4*x)*exp(4)*ln(x)+(3*x**4-12 
*x**3-4*x**2+17*x-1)*exp(4))/((ln(x)-x+4)*ln(-ln(x)+x-4)+x*ln(x)-x**2+4*x) 
,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x^{3} e^{4} - x e^{4} \log \left (x + \log \left (x - \log \left (x\right ) - 4\right )\right ) + 5 \, x e^{4} \] Input:

integrate((((-exp(4)*log(x)+(-4+x)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x 
)+(x^2-4*x)*exp(4))*log(log(-log(x)+x-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x 
^3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*x 
^4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^ 
2+4*x),x, algorithm="maxima")
 

Output:

-x^3*e^4 - x*e^4*log(x + log(x - log(x) - 4)) + 5*x*e^4
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x^{3} e^{4} - x e^{4} \log \left (x + \log \left (x - \log \left (x\right ) - 4\right )\right ) + 5 \, x e^{4} \] Input:

integrate((((-exp(4)*log(x)+(-4+x)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x 
)+(x^2-4*x)*exp(4))*log(log(-log(x)+x-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x 
^3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*x 
^4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^ 
2+4*x),x, algorithm="giac")
 

Output:

-x^3*e^4 - x*e^4*log(x + log(x - log(x) - 4)) + 5*x*e^4
 

Mupad [B] (verification not implemented)

Time = 1.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x\,{\mathrm {e}}^4\,\left (\ln \left (x+\ln \left (x-\ln \left (x\right )-4\right )\right )+x^2-5\right ) \] Input:

int(-(log(x + log(x - log(x) - 4))*(exp(4)*(4*x - x^2) - log(x - log(x) - 
4)*(exp(4)*(x - 4) - exp(4)*log(x)) + x*exp(4)*log(x)) + exp(4)*(4*x^2 - 1 
7*x + 12*x^3 - 3*x^4 + 1) + log(x - log(x) - 4)*(exp(4)*(5*x + 12*x^2 - 3* 
x^3 - 20) + exp(4)*log(x)*(3*x^2 - 5)) - exp(4)*log(x)*(4*x - 3*x^3))/(4*x 
 + log(x - log(x) - 4)*(log(x) - x + 4) + x*log(x) - x^2),x)
 

Output:

-x*exp(4)*(log(x + log(x - log(x) - 4)) + x^2 - 5)
 

Reduce [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=e^{4} x \left (-\mathrm {log}\left (\mathrm {log}\left (-\mathrm {log}\left (x \right )+x -4\right )+x \right )-x^{2}+5\right ) \] Input:

int((((-exp(4)*log(x)+(-4+x)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x)+(x^2 
-4*x)*exp(4))*log(log(-log(x)+x-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x^3-12* 
x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*x^4-12* 
x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^2+4*x) 
,x)
 

Output:

e**4*x*( - log(log( - log(x) + x - 4) + x) - x**2 + 5)