\(\int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} (e^{4+2 e^x x} (-1+e^x x)+(8 x+8 x^2) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} (-8 x-8 x^2+2 \log (5)+e^x (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5))))}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx\) [1624]

Optimal result

Integrand size = 208, antiderivative size = 38 \[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=\left (2-e^{-e^x+\frac {(2+2 x)^2}{e^{2+e^x x}-\log (5)}}\right ) x \] Output:

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

Mathematica [F]

\[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=\int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx \] Input:

Integrate[(2*E^(4 + 2*E^x*x) - 4*E^(2 + E^x*x)*Log[5] + 2*Log[5]^2 + E^((4 
 - E^(2 + x + E^x*x) + 8*x + 4*x^2 + E^x*Log[5])/(E^(2 + E^x*x) - Log[5])) 
*(E^(4 + 2*E^x*x)*(-1 + E^x*x) + (8*x + 8*x^2)*Log[5] - Log[5]^2 + E^x*x*L 
og[5]^2 + E^(2 + E^x*x)*(-8*x - 8*x^2 + 2*Log[5] + E^x*(4*x + 12*x^2 + 12* 
x^3 + 4*x^4 - 2*x*Log[5]))))/(E^(4 + 2*E^x*x) - 2*E^(2 + E^x*x)*Log[5] + L 
og[5]^2),x]
 

Output:

Integrate[(2*E^(4 + 2*E^x*x) - 4*E^(2 + E^x*x)*Log[5] + 2*Log[5]^2 + E^((4 
 - E^(2 + x + E^x*x) + 8*x + 4*x^2 + E^x*Log[5])/(E^(2 + E^x*x) - Log[5])) 
*(E^(4 + 2*E^x*x)*(-1 + E^x*x) + (8*x + 8*x^2)*Log[5] - Log[5]^2 + E^x*x*L 
og[5]^2 + E^(2 + E^x*x)*(-8*x - 8*x^2 + 2*Log[5] + E^x*(4*x + 12*x^2 + 12* 
x^3 + 4*x^4 - 2*x*Log[5]))))/(E^(4 + 2*E^x*x) - 2*E^(2 + E^x*x)*Log[5] + L 
og[5]^2), 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.

\(\displaystyle \int \frac {\left (\left (8 x^2+8 x\right ) \log (5)+e^{e^x x+2} \left (-8 x^2+e^x \left (4 x^4+12 x^3+12 x^2+4 x-2 x \log (5)\right )-8 x+2 \log (5)\right )+e^{2 e^x x+4} \left (e^x x-1\right )+e^x x \log ^2(5)-\log ^2(5)\right ) \exp \left (\frac {4 x^2+8 x-e^{e^x x+x+2}+e^x \log (5)+4}{e^{e^x x+2}-\log (5)}\right )+2 e^{2 e^x x+4}-4 e^{e^x x+2} \log (5)+2 \log ^2(5)}{e^{2 e^x x+4}-2 e^{e^x x+2} \log (5)+\log ^2(5)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (\left (8 x^2+8 x\right ) \log (5)+e^{e^x x+2} \left (-8 x^2+e^x \left (4 x^4+12 x^3+12 x^2+4 x-2 x \log (5)\right )-8 x+2 \log (5)\right )+e^{2 e^x x+4} \left (e^x x-1\right )+e^x x \log ^2(5)-\log ^2(5)\right ) \exp \left (\frac {4 x^2+8 x-e^{e^x x+x+2}+e^x \log (5)+4}{e^{e^x x+2}-\log (5)}\right )+2 e^{2 e^x x+4}-4 e^{e^x x+2} \log (5)+2 \log ^2(5)}{\left (e^{e^x x+2}-\log (5)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {5^{\frac {e^x}{e^{e^x x+2}-\log (5)}} \left (4 e^{e^x x+x+2} x^4+12 e^{e^x x+x+2} x^3-8 e^{e^x x+2} x^2+12 e^{e^x x+x+2} x^2+8 x^2 \log (5)-8 e^{e^x x+2} x+e^{2 e^x x+x+4} x-e^{2 e^x x+4}+e^x x \log ^2(5)+8 x \log (5)+4 e^{e^x x+x+2} x \left (1-\frac {\log (5)}{2}\right )+e^{e^x x+2} \log (25)-\log ^2(5)\right ) \exp \left (-\frac {-4 x^2-8 x+e^{e^x x+x+2}-4}{e^{e^x x+2}-\log (5)}\right )}{\left (e^{e^x x+2}-\log (5)\right )^2}+2\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {5^{\frac {e^x}{e^{e^x x+2}-\log (5)}} \left (4 e^{e^x x+x+2} x^4+12 e^{e^x x+x+2} x^3-8 e^{e^x x+2} x^2+12 e^{e^x x+x+2} x^2+8 x^2 \log (5)-8 e^{e^x x+2} x+e^{2 e^x x+x+4} x-e^{2 e^x x+4}+e^x x \log ^2(5)+8 x \log (5)+4 e^{e^x x+x+2} x \left (1-\frac {\log (5)}{2}\right )+e^{e^x x+2} \log (25)-\log ^2(5)\right ) \exp \left (-\frac {-4 x^2-8 x+e^{e^x x+x+2}-4}{e^{e^x x+2}-\log (5)}\right )}{\left (e^{e^x x+2}-\log (5)\right )^2}+2\right )dx\)

Input:

Int[(2*E^(4 + 2*E^x*x) - 4*E^(2 + E^x*x)*Log[5] + 2*Log[5]^2 + E^((4 - E^( 
2 + x + E^x*x) + 8*x + 4*x^2 + E^x*Log[5])/(E^(2 + E^x*x) - Log[5]))*(E^(4 
 + 2*E^x*x)*(-1 + E^x*x) + (8*x + 8*x^2)*Log[5] - Log[5]^2 + E^x*x*Log[5]^ 
2 + E^(2 + E^x*x)*(-8*x - 8*x^2 + 2*Log[5] + E^x*(4*x + 12*x^2 + 12*x^3 + 
4*x^4 - 2*x*Log[5]))))/(E^(4 + 2*E^x*x) - 2*E^(2 + E^x*x)*Log[5] + Log[5]^ 
2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 30.69 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32

method	result	size
risch	\(-x \,{\mathrm e}^{-\frac {-{\mathrm e}^{x +{\mathrm e}^{x} x +2}+{\mathrm e}^{x} \ln \left (5\right )+4 x^{2}+8 x +4}{-{\mathrm e}^{{\mathrm e}^{x} x +2}+\ln \left (5\right )}}+2 x\)	\(50\)
parallelrisch	\(-x \,{\mathrm e}^{-\frac {{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x} x +2}-{\mathrm e}^{x} \ln \left (5\right )-4 x^{2}-8 x -4}{{\mathrm e}^{{\mathrm e}^{x} x +2}-\ln \left (5\right )}}+2 x\)	\(51\)

Input:

int((((exp(x)*x-1)*exp(exp(x)*x+2)^2+((-2*x*ln(5)+4*x^4+12*x^3+12*x^2+4*x) 
*exp(x)+2*ln(5)-8*x^2-8*x)*exp(exp(x)*x+2)+x*ln(5)^2*exp(x)-ln(5)^2+(8*x^2 
+8*x)*ln(5))*exp((-exp(x)*exp(exp(x)*x+2)+exp(x)*ln(5)+4*x^2+8*x+4)/(exp(e 
xp(x)*x+2)-ln(5)))+2*exp(exp(x)*x+2)^2-4*ln(5)*exp(exp(x)*x+2)+2*ln(5)^2)/ 
(exp(exp(x)*x+2)^2-2*ln(5)*exp(exp(x)*x+2)+ln(5)^2),x,method=_RETURNVERBOS 
E)
 

Output:

-x*exp(-(-exp(x+exp(x)*x+2)+exp(x)*ln(5)+4*x^2+8*x+4)/(-exp(exp(x)*x+2)+ln 
(5)))+2*x
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=-x e^{\left (-\frac {4 \, {\left (x^{2} + 2 \, x + 1\right )} e^{x} + e^{\left (2 \, x\right )} \log \left (5\right ) - e^{\left (x e^{x} + 2 \, x + 2\right )}}{e^{x} \log \left (5\right ) - e^{\left (x e^{x} + x + 2\right )}}\right )} + 2 \, x \] Input:

integrate((((exp(x)*x-1)*exp(exp(x)*x+2)^2+((-2*x*log(5)+4*x^4+12*x^3+12*x 
^2+4*x)*exp(x)+2*log(5)-8*x^2-8*x)*exp(exp(x)*x+2)+x*log(5)^2*exp(x)-log(5 
)^2+(8*x^2+8*x)*log(5))*exp((-exp(x)*exp(exp(x)*x+2)+exp(x)*log(5)+4*x^2+8 
*x+4)/(exp(exp(x)*x+2)-log(5)))+2*exp(exp(x)*x+2)^2-4*log(5)*exp(exp(x)*x+ 
2)+2*log(5)^2)/(exp(exp(x)*x+2)^2-2*log(5)*exp(exp(x)*x+2)+log(5)^2),x, al 
gorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 3.85 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=- x e^{\frac {4 x^{2} + 8 x - e^{x} e^{x e^{x} + 2} + e^{x} \log {\left (5 \right )} + 4}{e^{x e^{x} + 2} - \log {\left (5 \right )}}} + 2 x \] Input:

integrate((((exp(x)*x-1)*exp(exp(x)*x+2)**2+((-2*x*ln(5)+4*x**4+12*x**3+12 
*x**2+4*x)*exp(x)+2*ln(5)-8*x**2-8*x)*exp(exp(x)*x+2)+x*ln(5)**2*exp(x)-ln 
(5)**2+(8*x**2+8*x)*ln(5))*exp((-exp(x)*exp(exp(x)*x+2)+exp(x)*ln(5)+4*x** 
2+8*x+4)/(exp(exp(x)*x+2)-ln(5)))+2*exp(exp(x)*x+2)**2-4*ln(5)*exp(exp(x)* 
x+2)+2*ln(5)**2)/(exp(exp(x)*x+2)**2-2*ln(5)*exp(exp(x)*x+2)+ln(5)**2),x)
 

Output:

-x*exp((4*x**2 + 8*x - exp(x)*exp(x*exp(x) + 2) + exp(x)*log(5) + 4)/(exp( 
x*exp(x) + 2) - log(5))) + 2*x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (32) = 64\).

Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.74 \[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=-x e^{\left (\frac {4 \, x^{2}}{e^{\left (x e^{x} + 2\right )} - \log \left (5\right )} + \frac {e^{x} \log \left (5\right )}{e^{\left (x e^{x} + 2\right )} - \log \left (5\right )} + \frac {8 \, x}{e^{\left (x e^{x} + 2\right )} - \log \left (5\right )} - \frac {e^{\left (x e^{x} + x + 2\right )}}{e^{\left (x e^{x} + 2\right )} - \log \left (5\right )} + \frac {4}{e^{\left (x e^{x} + 2\right )} - \log \left (5\right )}\right )} + 2 \, x \] Input:

integrate((((exp(x)*x-1)*exp(exp(x)*x+2)^2+((-2*x*log(5)+4*x^4+12*x^3+12*x 
^2+4*x)*exp(x)+2*log(5)-8*x^2-8*x)*exp(exp(x)*x+2)+x*log(5)^2*exp(x)-log(5 
)^2+(8*x^2+8*x)*log(5))*exp((-exp(x)*exp(exp(x)*x+2)+exp(x)*log(5)+4*x^2+8 
*x+4)/(exp(exp(x)*x+2)-log(5)))+2*exp(exp(x)*x+2)^2-4*log(5)*exp(exp(x)*x+ 
2)+2*log(5)^2)/(exp(exp(x)*x+2)^2-2*log(5)*exp(exp(x)*x+2)+log(5)^2),x, al 
gorithm="maxima")
 

Output:

-x*e^(4*x^2/(e^(x*e^x + 2) - log(5)) + e^x*log(5)/(e^(x*e^x + 2) - log(5)) 
 + 8*x/(e^(x*e^x + 2) - log(5)) - e^(x*e^x + x + 2)/(e^(x*e^x + 2) - log(5 
)) + 4/(e^(x*e^x + 2) - log(5))) + 2*x
 

Giac [F]

\[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=\int { -\frac {{\left (x e^{x} \log \left (5\right )^{2} + {\left (x e^{x} - 1\right )} e^{\left (2 \, x e^{x} + 4\right )} - 2 \, {\left (4 \, x^{2} - {\left (2 \, x^{4} + 6 \, x^{3} + 6 \, x^{2} - x \log \left (5\right ) + 2 \, x\right )} e^{x} + 4 \, x - \log \left (5\right )\right )} e^{\left (x e^{x} + 2\right )} + 8 \, {\left (x^{2} + x\right )} \log \left (5\right ) - \log \left (5\right )^{2}\right )} e^{\left (\frac {4 \, x^{2} + e^{x} \log \left (5\right ) + 8 \, x - e^{\left (x e^{x} + x + 2\right )} + 4}{e^{\left (x e^{x} + 2\right )} - \log \left (5\right )}\right )} - 4 \, e^{\left (x e^{x} + 2\right )} \log \left (5\right ) + 2 \, \log \left (5\right )^{2} + 2 \, e^{\left (2 \, x e^{x} + 4\right )}}{2 \, e^{\left (x e^{x} + 2\right )} \log \left (5\right ) - \log \left (5\right )^{2} - e^{\left (2 \, x e^{x} + 4\right )}} \,d x } \] Input:

integrate((((exp(x)*x-1)*exp(exp(x)*x+2)^2+((-2*x*log(5)+4*x^4+12*x^3+12*x 
^2+4*x)*exp(x)+2*log(5)-8*x^2-8*x)*exp(exp(x)*x+2)+x*log(5)^2*exp(x)-log(5 
)^2+(8*x^2+8*x)*log(5))*exp((-exp(x)*exp(exp(x)*x+2)+exp(x)*log(5)+4*x^2+8 
*x+4)/(exp(exp(x)*x+2)-log(5)))+2*exp(exp(x)*x+2)^2-4*log(5)*exp(exp(x)*x+ 
2)+2*log(5)^2)/(exp(exp(x)*x+2)^2-2*log(5)*exp(exp(x)*x+2)+log(5)^2),x, al 
gorithm="giac")
 

Output:

integrate(-((x*e^x*log(5)^2 + (x*e^x - 1)*e^(2*x*e^x + 4) - 2*(4*x^2 - (2* 
x^4 + 6*x^3 + 6*x^2 - x*log(5) + 2*x)*e^x + 4*x - log(5))*e^(x*e^x + 2) + 
8*(x^2 + x)*log(5) - log(5)^2)*e^((4*x^2 + e^x*log(5) + 8*x - e^(x*e^x + x 
 + 2) + 4)/(e^(x*e^x + 2) - log(5))) - 4*e^(x*e^x + 2)*log(5) + 2*log(5)^2 
 + 2*e^(2*x*e^x + 4))/(2*e^(x*e^x + 2)*log(5) - log(5)^2 - e^(2*x*e^x + 4) 
), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=\int \frac {2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x+4}-4\,{\mathrm {e}}^{x\,{\mathrm {e}}^x+2}\,\ln \left (5\right )+{\mathrm {e}}^{-\frac {8\,x+{\mathrm {e}}^x\,\ln \left (5\right )-{\mathrm {e}}^{x\,{\mathrm {e}}^x+2}\,{\mathrm {e}}^x+4\,x^2+4}{\ln \left (5\right )-{\mathrm {e}}^{x\,{\mathrm {e}}^x+2}}}\,\left (\ln \left (5\right )\,\left (8\,x^2+8\,x\right )-{\mathrm {e}}^{x\,{\mathrm {e}}^x+2}\,\left (8\,x-2\,\ln \left (5\right )-{\mathrm {e}}^x\,\left (4\,x-2\,x\,\ln \left (5\right )+12\,x^2+12\,x^3+4\,x^4\right )+8\,x^2\right )-{\ln \left (5\right )}^2+{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x+4}\,\left (x\,{\mathrm {e}}^x-1\right )+x\,{\mathrm {e}}^x\,{\ln \left (5\right )}^2\right )+2\,{\ln \left (5\right )}^2}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x+4}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^x+2}\,\ln \left (5\right )+{\ln \left (5\right )}^2} \,d x \] Input:

int((2*exp(2*x*exp(x) + 4) - 4*exp(x*exp(x) + 2)*log(5) + exp(-(8*x + exp( 
x)*log(5) - exp(x*exp(x) + 2)*exp(x) + 4*x^2 + 4)/(log(5) - exp(x*exp(x) + 
 2)))*(log(5)*(8*x + 8*x^2) - exp(x*exp(x) + 2)*(8*x - 2*log(5) - exp(x)*( 
4*x - 2*x*log(5) + 12*x^2 + 12*x^3 + 4*x^4) + 8*x^2) - log(5)^2 + exp(2*x* 
exp(x) + 4)*(x*exp(x) - 1) + x*exp(x)*log(5)^2) + 2*log(5)^2)/(exp(2*x*exp 
(x) + 4) - 2*exp(x*exp(x) + 2)*log(5) + log(5)^2),x)
 

Output:

int((2*exp(2*x*exp(x) + 4) - 4*exp(x*exp(x) + 2)*log(5) + exp(-(8*x + exp( 
x)*log(5) - exp(x*exp(x) + 2)*exp(x) + 4*x^2 + 4)/(log(5) - exp(x*exp(x) + 
 2)))*(log(5)*(8*x + 8*x^2) - exp(x*exp(x) + 2)*(8*x - 2*log(5) - exp(x)*( 
4*x - 2*x*log(5) + 12*x^2 + 12*x^3 + 4*x^4) + 8*x^2) - log(5)^2 + exp(2*x* 
exp(x) + 4)*(x*exp(x) - 1) + x*exp(x)*log(5)^2) + 2*log(5)^2)/(exp(2*x*exp 
(x) + 4) - 2*exp(x*exp(x) + 2)*log(5) + log(5)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {2 e^{4+2 e^x x}-4 e^{2+e^x x} \log (5)+2 \log ^2(5)+e^{\frac {4-e^{2+x+e^x x}+8 x+4 x^2+e^x \log (5)}{e^{2+e^x x}-\log (5)}} \left (e^{4+2 e^x x} \left (-1+e^x x\right )+\left (8 x+8 x^2\right ) \log (5)-\log ^2(5)+e^x x \log ^2(5)+e^{2+e^x x} \left (-8 x-8 x^2+2 \log (5)+e^x \left (4 x+12 x^2+12 x^3+4 x^4-2 x \log (5)\right )\right )\right )}{e^{4+2 e^x x}-2 e^{2+e^x x} \log (5)+\log ^2(5)} \, dx=\frac {x \left (2 e^{e^{x}}-e^{\frac {4 x^{2}+8 x +4}{e^{e^{x} x} e^{2}-\mathrm {log}\left (5\right )}}\right )}{e^{e^{x}}} \] Input:

int((((exp(x)*x-1)*exp(exp(x)*x+2)^2+((-2*x*log(5)+4*x^4+12*x^3+12*x^2+4*x 
)*exp(x)+2*log(5)-8*x^2-8*x)*exp(exp(x)*x+2)+x*log(5)^2*exp(x)-log(5)^2+(8 
*x^2+8*x)*log(5))*exp((-exp(x)*exp(exp(x)*x+2)+exp(x)*log(5)+4*x^2+8*x+4)/ 
(exp(exp(x)*x+2)-log(5)))+2*exp(exp(x)*x+2)^2-4*log(5)*exp(exp(x)*x+2)+2*l 
og(5)^2)/(exp(exp(x)*x+2)^2-2*log(5)*exp(exp(x)*x+2)+log(5)^2),x)
 

Output:

(x*(2*e**(e**x) - e**((4*x**2 + 8*x + 4)/(e**(e**x*x)*e**2 - log(5)))))/e* 
*(e**x)