\(\int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5)+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 (256-32 x^2+x^4)}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx\) [164]

Optimal result

Integrand size = 138, antiderivative size = 29 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x}{e \left (e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)\right )} \] Output:

x/exp(1)/(exp((x^2-16)^2/exp(4))-ln(4-x))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x}{e \left (e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)\right )} \] Input:

Integrate[(E^4*x + E^((256 - 32*x^2 + x^4)/E^4)*(E^4*(-4 + x) - 256*x^2 + 
64*x^3 + 16*x^4 - 4*x^5) + E^4*(4 - x)*Log[4 - x])/(E^(5 + (2*(256 - 32*x^ 
2 + x^4))/E^4)*(-4 + x) + E^(5 + (256 - 32*x^2 + x^4)/E^4)*(8 - 2*x)*Log[4 
 - x] + E^5*(-4 + x)*Log[4 - x]^2),x]
 

Output:

x/(E*(E^((-16 + x^2)^2/E^4) - Log[4 - 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.

\(\displaystyle \int \frac {e^{\frac {x^4-32 x^2+256}{e^4}} \left (-4 x^5+16 x^4+64 x^3-256 x^2+e^4 (x-4)\right )+e^4 x+e^4 (4-x) \log (4-x)}{e^{\frac {2 \left (x^4-32 x^2+256\right )}{e^4}+5} (x-4)+e^{\frac {x^4-32 x^2+256}{e^4}+5} (8-2 x) \log (4-x)+e^5 (x-4) \log ^2(4-x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{\frac {64 x^2}{e^4}-5} \left (-e^{\frac {x^4-32 x^2+256}{e^4}} \left (-4 x^5+16 x^4+64 x^3-256 x^2+e^4 (x-4)\right )-e^4 x-e^4 (4-x) \log (4-x)\right )}{(4-x) \left (e^{\frac {x^4}{e^4}+\frac {256}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{\frac {64 x^2}{e^4}-5} \left (-4 x^4+64 x^2+e^4\right )}{e^{\frac {\left (x^2+16\right )^2}{e^4}}-e^{\frac {64 x^2}{e^4}} \log (4-x)}-\frac {e^{\frac {64 x^2}{e^4}-5} x \left (4 x^4 \log (4-x)-16 x^3 \log (4-x)-64 x^2 \log (4-x)+256 x \log (4-x)-e^4\right )}{(x-4) \left (e^{\frac {x^4}{e^4}+\frac {256}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-4 x^4+64 x^2+e^4}{e^5 \left (e^{\frac {\left (x^2-16\right )^2}{e^4}}-\log (4-x)\right )}+\frac {e^{\frac {64 x^2}{e^4}-5} x \left (-4 x^4 \log (4-x)+16 x^3 \log (4-x)+64 x^2 \log (4-x)-256 x \log (4-x)+e^4\right )}{(x-4) \left (e^{\frac {32 x^2}{e^4}} \log (4-x)-e^{\frac {x^4}{e^4}+\frac {256}{e^4}}\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {-4 x^4+64 x^2+e^4}{e^5 \left (e^{\frac {\left (x^2-16\right )^2}{e^4}}-\log (4-x)\right )}+\frac {e^{\frac {64 x^2}{e^4}-5} x \left (-4 x^4 \log (4-x)+16 x^3 \log (4-x)+64 x^2 \log (4-x)-256 x \log (4-x)+e^4\right )}{(x-4) \left (e^{\frac {32 x^2}{e^4}} \log (4-x)-e^{\frac {x^4}{e^4}+\frac {256}{e^4}}\right )^2}\right )dx\)

Input:

Int[(E^4*x + E^((256 - 32*x^2 + x^4)/E^4)*(E^4*(-4 + x) - 256*x^2 + 64*x^3 
 + 16*x^4 - 4*x^5) + E^4*(4 - x)*Log[4 - x])/(E^(5 + (2*(256 - 32*x^2 + x^ 
4))/E^4)*(-4 + x) + E^(5 + (256 - 32*x^2 + x^4)/E^4)*(8 - 2*x)*Log[4 - x] 
+ E^5*(-4 + x)*Log[4 - x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {{\mathrm e}^{-1} x}{{\mathrm e}^{\left (x -4\right )^{2} \left (4+x \right )^{2} {\mathrm e}^{-4}}-\ln \left (-x +4\right )}\)	\(30\)
parallelrisch	\(-\frac {x \,{\mathrm e}^{-1}}{\ln \left (-x +4\right )-{\mathrm e}^{\left (x^{4}-32 x^{2}+256\right ) {\mathrm e}^{-4}}}\)	\(35\)

Input:

int(((-x+4)*exp(4)*ln(-x+4)+((x-4)*exp(4)-4*x^5+16*x^4+64*x^3-256*x^2)*exp 
((x^4-32*x^2+256)/exp(4))+x*exp(4))/((x-4)*exp(1)*exp(4)*ln(-x+4)^2+(-2*x+ 
8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*ln(-x+4)+(x-4)*exp(1)*exp(4) 
*exp((x^4-32*x^2+256)/exp(4))^2),x,method=_RETURNVERBOSE)
 

Output:

exp(-1)*x/(exp((x-4)^2*(4+x)^2*exp(-4))-ln(-x+4))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=-\frac {x e^{4}}{e^{5} \log \left (-x + 4\right ) - e^{\left ({\left (x^{4} - 32 \, x^{2} + 5 \, e^{4} + 256\right )} e^{\left (-4\right )}\right )}} \] Input:

integrate(((-x+4)*exp(4)*log(-x+4)+((-4+x)*exp(4)-4*x^5+16*x^4+64*x^3-256* 
x^2)*exp((x^4-32*x^2+256)/exp(4))+x*exp(4))/((-4+x)*exp(1)*exp(4)*log(-x+4 
)^2+(-2*x+8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*log(-x+4)+(-4+x)*e 
xp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))^2),x, algorithm="fricas")
 

Output:

-x*e^4/(e^5*log(-x + 4) - e^((x^4 - 32*x^2 + 5*e^4 + 256)*e^(-4)))
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x}{e e^{\frac {x^{4} - 32 x^{2} + 256}{e^{4}}} - e \log {\left (4 - x \right )}} \] Input:

integrate(((-x+4)*exp(4)*ln(-x+4)+((-4+x)*exp(4)-4*x**5+16*x**4+64*x**3-25 
6*x**2)*exp((x**4-32*x**2+256)/exp(4))+x*exp(4))/((-4+x)*exp(1)*exp(4)*ln( 
-x+4)**2+(-2*x+8)*exp(1)*exp(4)*exp((x**4-32*x**2+256)/exp(4))*ln(-x+4)+(- 
4+x)*exp(1)*exp(4)*exp((x**4-32*x**2+256)/exp(4))**2),x)
 

Output:

x/(E*exp((x**4 - 32*x**2 + 256)*exp(-4)) - E*log(4 - x))
 

Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=-\frac {x e^{\left (32 \, x^{2} e^{\left (-4\right )}\right )}}{e^{\left (32 \, x^{2} e^{\left (-4\right )} + 1\right )} \log \left (-x + 4\right ) - e^{\left (x^{4} e^{\left (-4\right )} + 256 \, e^{\left (-4\right )} + 1\right )}} \] Input:

integrate(((-x+4)*exp(4)*log(-x+4)+((-4+x)*exp(4)-4*x^5+16*x^4+64*x^3-256* 
x^2)*exp((x^4-32*x^2+256)/exp(4))+x*exp(4))/((-4+x)*exp(1)*exp(4)*log(-x+4 
)^2+(-2*x+8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*log(-x+4)+(-4+x)*e 
xp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))^2),x, algorithm="maxima")
 

Output:

-x*e^(32*x^2*e^(-4))/(e^(32*x^2*e^(-4) + 1)*log(-x + 4) - e^(x^4*e^(-4) + 
256*e^(-4) + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (26) = 52\).

Time = 1.08 (sec) , antiderivative size = 914, normalized size of antiderivative = 31.52 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\text {Too large to display} \] Input:

integrate(((-x+4)*exp(4)*log(-x+4)+((-4+x)*exp(4)-4*x^5+16*x^4+64*x^3-256* 
x^2)*exp((x^4-32*x^2+256)/exp(4))+x*exp(4))/((-4+x)*exp(1)*exp(4)*log(-x+4 
)^2+(-2*x+8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*log(-x+4)+(-4+x)*e 
xp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))^2),x, algorithm="giac")
 

Output:

-(4*x^5*e^(x^4*e^(-4))*log(-x + 4)^2 - 4*x^5*e^(x^4*e^(-4) + (x^4 - 32*x^2 
)*e^(-4) + 256*e^(-4))*log(-x + 4) - 16*x^4*e^(x^4*e^(-4))*log(-x + 4)^2 + 
 16*x^4*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4))*log(-x + 4) - 
64*x^3*e^(x^4*e^(-4))*log(-x + 4)^2 + 64*x^3*e^(x^4*e^(-4) + (x^4 - 32*x^2 
)*e^(-4) + 256*e^(-4))*log(-x + 4) + 256*x^2*e^(x^4*e^(-4))*log(-x + 4)^2 
- 256*x^2*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4))*log(-x + 4) 
- x*e^(x^4*e^(-4) + 4)*log(-x + 4) + x*e^(2*(x^4 - 16*x^2)*e^(-4) + 256*e^ 
(-4) + 4))/(4*x^4*e^(x^4*e^(-4) + 1)*log(-x + 4)^3 - 4*x^4*e^(x^4*e^(-4) + 
 (x^4 - 32*x^2)*e^(-4) + 256*e^(-4) + 1)*log(-x + 4)^2 - 4*x^4*e^(2*(x^4 - 
 16*x^2)*e^(-4) + 256*e^(-4) + 1)*log(-x + 4)^2 - 16*x^3*e^(x^4*e^(-4) + 1 
)*log(-x + 4)^3 + 4*x^4*e^(2*(x^4 - 16*x^2)*e^(-4) + (x^4 - 32*x^2)*e^(-4) 
 + 512*e^(-4) + 1)*log(-x + 4) + 16*x^3*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^( 
-4) + 256*e^(-4) + 1)*log(-x + 4)^2 + 16*x^3*e^(2*(x^4 - 16*x^2)*e^(-4) + 
256*e^(-4) + 1)*log(-x + 4)^2 - 64*x^2*e^(x^4*e^(-4) + 1)*log(-x + 4)^3 - 
16*x^3*e^(2*(x^4 - 16*x^2)*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 512*e^(-4) + 1 
)*log(-x + 4) + 64*x^2*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4) 
+ 1)*log(-x + 4)^2 + 64*x^2*e^(2*(x^4 - 16*x^2)*e^(-4) + 256*e^(-4) + 1)*l 
og(-x + 4)^2 + 256*x*e^(x^4*e^(-4) + 1)*log(-x + 4)^3 - 64*x^2*e^(2*(x^4 - 
 16*x^2)*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 512*e^(-4) + 1)*log(-x + 4) - 25 
6*x*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4) + 1)*log(-x + 4)...
 

Mupad [B] (verification not implemented)

Time = 2.85 (sec) , antiderivative size = 158, normalized size of antiderivative = 5.45 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x\,{\mathrm {e}}^{-1}\,{\left (4\,{\mathrm {e}}^4-x\,{\mathrm {e}}^4\right )}^2-{\mathrm {e}}^{-5}\,\ln \left (4-x\right )\,{\left (4\,{\mathrm {e}}^4-x\,{\mathrm {e}}^4\right )}^2\,\left (4\,x^5-16\,x^4-64\,x^3+256\,x^2\right )}{\left (\ln \left (4-x\right )-{\mathrm {e}}^{{\mathrm {e}}^{-4}\,x^4-32\,{\mathrm {e}}^{-4}\,x^2+256\,{\mathrm {e}}^{-4}}\right )\,\left (x-4\right )\,\left (4\,{\mathrm {e}}^8-x\,{\mathrm {e}}^8+512\,x^2\,{\mathrm {e}}^4\,\ln \left (4-x\right )-32\,x^4\,{\mathrm {e}}^4\,\ln \left (4-x\right )+4\,x^5\,{\mathrm {e}}^4\,\ln \left (4-x\right )-1024\,x\,{\mathrm {e}}^4\,\ln \left (4-x\right )\right )} \] Input:

int((x*exp(4) + exp(exp(-4)*(x^4 - 32*x^2 + 256))*(exp(4)*(x - 4) - 256*x^ 
2 + 64*x^3 + 16*x^4 - 4*x^5) - exp(4)*log(4 - x)*(x - 4))/(exp(5)*exp(2*ex 
p(-4)*(x^4 - 32*x^2 + 256))*(x - 4) + exp(5)*log(4 - x)^2*(x - 4) - exp(5) 
*exp(exp(-4)*(x^4 - 32*x^2 + 256))*log(4 - x)*(2*x - 8)),x)
 

Output:

(x*exp(-1)*(4*exp(4) - x*exp(4))^2 - exp(-5)*log(4 - x)*(4*exp(4) - x*exp( 
4))^2*(256*x^2 - 64*x^3 - 16*x^4 + 4*x^5))/((log(4 - x) - exp(256*exp(-4) 
- 32*x^2*exp(-4) + x^4*exp(-4)))*(x - 4)*(4*exp(8) - x*exp(8) + 512*x^2*ex 
p(4)*log(4 - x) - 32*x^4*exp(4)*log(4 - x) + 4*x^5*exp(4)*log(4 - x) - 102 
4*x*exp(4)*log(4 - x)))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {e^{\frac {32 x^{2}}{e^{4}}} x}{e \left (e^{\frac {x^{4}+256}{e^{4}}}-e^{\frac {32 x^{2}}{e^{4}}} \mathrm {log}\left (-x +4\right )\right )} \] Input:

int(((-x+4)*exp(4)*log(-x+4)+((-4+x)*exp(4)-4*x^5+16*x^4+64*x^3-256*x^2)*e 
xp((x^4-32*x^2+256)/exp(4))+x*exp(4))/((-4+x)*exp(1)*exp(4)*log(-x+4)^2+(- 
2*x+8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*log(-x+4)+(-4+x)*exp(1)* 
exp(4)*exp((x^4-32*x^2+256)/exp(4))^2),x)
 

Output:

(e**((32*x**2)/e**4)*x)/(e*(e**((x**4 + 256)/e**4) - e**((32*x**2)/e**4)*l 
og( - x + 4)))