\(\int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} (-2 e^5+e^{9-4 x+x^2} (-8+4 x))}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} (121+88 x+16 x^2)} \, dx\) [2026]

Optimal result

Integrand size = 106, antiderivative size = 28 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=\frac {1}{5-2 \left (e^{-5+e^{(2-x)^2}-x}+2 (4+x)\right )} \] Output:

1/(-11-4*x-2*exp(exp((2-x)^2)-x)/exp(5))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=-\frac {e^{5+x}}{2 e^{e^{(-2+x)^2}}+e^{5+x} (11+4 x)} \] Input:

Integrate[(4*E^10 + E^(E^(4 - 4*x + x^2) - x)*(-2*E^5 + E^(9 - 4*x + x^2)* 
(-8 + 4*x)))/(4*E^(2*E^(4 - 4*x + x^2) - 2*x) + E^(5 + E^(4 - 4*x + x^2) - 
 x)*(44 + 16*x) + E^10*(121 + 88*x + 16*x^2)),x]
 

Output:

-(E^(5 + x)/(2*E^E^(-2 + x)^2 + E^(5 + x)*(11 + 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.

\(\displaystyle \int \frac {e^{e^{x^2-4 x+4}-x} \left (e^{x^2-4 x+9} (4 x-8)-2 e^5\right )+4 e^{10}}{e^{e^{x^2-4 x+4}-x+5} (16 x+44)+4 e^{2 e^{x^2-4 x+4}-2 x}+e^{10} \left (16 x^2+88 x+121\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{2 x} \left (e^{e^{x^2-4 x+4}-x} \left (e^{x^2-4 x+9} (4 x-8)-2 e^5\right )+4 e^{10}\right )}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 e^{x^2-3 x+e^{(x-2)^2}+9} (x-2)}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}+\frac {2 e^{x+5} \left (2 e^{x+5}-e^{e^{(x-2)^2}}\right )}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -8 \int \frac {e^{x^2-3 x+e^{(x-2)^2}+9}}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx+4 \int \frac {e^{x^2-3 x+e^{(x-2)^2}+9} x}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx-2 \int \frac {e^{x+e^{(x-2)^2}+5}}{\left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx-8 \int \frac {e^{x+e^{(x-2)^2}+5}}{(4 x+11) \left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )^2}dx+4 \int \frac {e^{x+5}}{(4 x+11) \left (4 e^{x+5} x+2 e^{e^{(x-2)^2}}+11 e^{x+5}\right )}dx\)

Input:

Int[(4*E^10 + E^(E^(4 - 4*x + x^2) - x)*(-2*E^5 + E^(9 - 4*x + x^2)*(-8 + 
4*x)))/(4*E^(2*E^(4 - 4*x + x^2) - 2*x) + E^(5 + E^(4 - 4*x + x^2) - x)*(4 
4 + 16*x) + E^10*(121 + 88*x + 16*x^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
risch	\(-\frac {{\mathrm e}^{5}}{4 x \,{\mathrm e}^{5}+11 \,{\mathrm e}^{5}+2 \,{\mathrm e}^{{\mathrm e}^{\left (-2+x \right )^{2}}-x}}\)	\(30\)
norman	\(-\frac {{\mathrm e}^{5}}{4 x \,{\mathrm e}^{5}+11 \,{\mathrm e}^{5}+2 \,{\mathrm e}^{{\mathrm e}^{x^{2}-4 x +4}-x}}\)	\(33\)
parallelrisch	\(-\frac {{\mathrm e}^{5}}{4 x \,{\mathrm e}^{5}+11 \,{\mathrm e}^{5}+2 \,{\mathrm e}^{{\mathrm e}^{x^{2}-4 x +4}-x}}\)	\(33\)

Input:

int((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+4*exp( 
5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4)-x)+(1 
6*x^2+88*x+121)*exp(5)^2),x,method=_RETURNVERBOSE)
 

Output:

-exp(5)/(4*x*exp(5)+11*exp(5)+2*exp(exp((-2+x)^2)-x))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=-\frac {e^{10}}{{\left (4 \, x + 11\right )} e^{10} + 2 \, e^{\left (-{\left ({\left (x - 5\right )} e^{5} - e^{\left (x^{2} - 4 \, x + 9\right )}\right )} e^{\left (-5\right )}\right )}} \] Input:

integrate((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+ 
4*exp(5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4) 
-x)+(16*x^2+88*x+121)*exp(5)^2),x, algorithm="fricas")
 

Output:

-e^10/((4*x + 11)*e^10 + 2*e^(-((x - 5)*e^5 - e^(x^2 - 4*x + 9))*e^(-5)))
 

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=- \frac {e^{5}}{4 x e^{5} + 2 e^{- x + e^{x^{2} - 4 x + 4}} + 11 e^{5}} \] Input:

integrate((((4*x-8)*exp(5)*exp(x**2-4*x+4)-2*exp(5))*exp(exp(x**2-4*x+4)-x 
)+4*exp(5)**2)/(4*exp(exp(x**2-4*x+4)-x)**2+(16*x+44)*exp(5)*exp(exp(x**2- 
4*x+4)-x)+(16*x**2+88*x+121)*exp(5)**2),x)
 

Output:

-exp(5)/(4*x*exp(5) + 2*exp(-x + exp(x**2 - 4*x + 4)) + 11*exp(5))
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=-\frac {e^{\left (x + 5\right )}}{{\left (4 \, x e^{5} + 11 \, e^{5}\right )} e^{x} + 2 \, e^{\left (e^{\left (x^{2} - 4 \, x + 4\right )}\right )}} \] Input:

integrate((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+ 
4*exp(5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4) 
-x)+(16*x^2+88*x+121)*exp(5)^2),x, algorithm="maxima")
 

Output:

-e^(x + 5)/((4*x*e^5 + 11*e^5)*e^x + 2*e^(e^(x^2 - 4*x + 4)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2593 vs. \(2 (23) = 46\).

Time = 0.22 (sec) , antiderivative size = 2593, normalized size of antiderivative = 92.61 \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+ 
4*exp(5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4) 
-x)+(16*x^2+88*x+121)*exp(5)^2),x, algorithm="giac")
 

Output:

-(1024*x^6*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) + 7168*x^5*e^(2*x^2 
+ 3/2*x + e^(x^2 - 4*x + 4) + 23) + 1024*x^5*e^(2*x^2 + 1/2*x + 2*e^(x^2 - 
 4*x + 4) + 18) - 1024*x^5*e^(x^2 + 11/2*x + e^(x^2 - 4*x + 4) + 19) + 550 
4*x^4*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) + 4352*x^4*e^(2*x^2 + 1/2 
*x + 2*e^(x^2 - 4*x + 4) + 18) + 256*x^4*e^(2*x^2 - 1/2*x + 3*e^(x^2 - 4*x 
 + 4) + 13) - 10240*x^4*e^(x^2 + 11/2*x + e^(x^2 - 4*x + 4) + 19) - 1024*x 
^4*e^(x^2 + 9/2*x + 2*e^(x^2 - 4*x + 4) + 14) + 256*x^4*e^(19/2*x + e^(x^2 
 - 4*x + 4) + 15) - 55616*x^3*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) - 
 6464*x^3*e^(2*x^2 + 1/2*x + 2*e^(x^2 - 4*x + 4) + 18) + 384*x^3*e^(2*x^2 
- 1/2*x + 3*e^(x^2 - 4*x + 4) + 13) - 30336*x^3*e^(x^2 + 11/2*x + e^(x^2 - 
 4*x + 4) + 19) - 7424*x^3*e^(x^2 + 9/2*x + 2*e^(x^2 - 4*x + 4) + 14) - 25 
6*x^3*e^(x^2 + 7/2*x + 3*e^(x^2 - 4*x + 4) + 9) + 3328*x^3*e^(19/2*x + e^( 
x^2 - 4*x + 4) + 15) + 256*x^3*e^(17/2*x + 2*e^(x^2 - 4*x + 4) + 10) - 963 
16*x^2*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) + 23) - 37840*x^2*e^(2*x^2 + 1 
/2*x + 2*e^(x^2 - 4*x + 4) + 18) - 2672*x^2*e^(2*x^2 - 1/2*x + 3*e^(x^2 - 
4*x + 4) + 13) + 1408*x^2*e^(x^2 + 11/2*x + e^(x^2 - 4*x + 4) + 19) - 9920 
*x^2*e^(x^2 + 9/2*x + 2*e^(x^2 - 4*x + 4) + 14) - 1152*x^2*e^(x^2 + 7/2*x 
+ 3*e^(x^2 - 4*x + 4) + 9) + 16096*x^2*e^(19/2*x + e^(x^2 - 4*x + 4) + 15) 
 + 2624*x^2*e^(17/2*x + 2*e^(x^2 - 4*x + 4) + 10) + 64*x^2*e^(15/2*x + 3*e 
^(x^2 - 4*x + 4) + 5) + 106480*x*e^(2*x^2 + 3/2*x + e^(x^2 - 4*x + 4) +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=\int \frac {4\,{\mathrm {e}}^{10}-{\mathrm {e}}^{{\mathrm {e}}^{x^2-4\,x+4}-x}\,\left (2\,{\mathrm {e}}^5-{\mathrm {e}}^5\,{\mathrm {e}}^{x^2-4\,x+4}\,\left (4\,x-8\right )\right )}{4\,{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2-4\,x+4}-2\,x}+{\mathrm {e}}^{10}\,\left (16\,x^2+88\,x+121\right )+{\mathrm {e}}^{{\mathrm {e}}^{x^2-4\,x+4}-x}\,{\mathrm {e}}^5\,\left (16\,x+44\right )} \,d x \] Input:

int((4*exp(10) - exp(exp(x^2 - 4*x + 4) - x)*(2*exp(5) - exp(5)*exp(x^2 - 
4*x + 4)*(4*x - 8)))/(4*exp(2*exp(x^2 - 4*x + 4) - 2*x) + exp(10)*(88*x + 
16*x^2 + 121) + exp(exp(x^2 - 4*x + 4) - x)*exp(5)*(16*x + 44)),x)
 

Output:

int((4*exp(10) - exp(exp(x^2 - 4*x + 4) - x)*(2*exp(5) - exp(5)*exp(x^2 - 
4*x + 4)*(4*x - 8)))/(4*exp(2*exp(x^2 - 4*x + 4) - 2*x) + exp(10)*(88*x + 
16*x^2 + 121) + exp(exp(x^2 - 4*x + 4) - x)*exp(5)*(16*x + 44)), x)
 

Reduce [F]

\[ \int \frac {4 e^{10}+e^{e^{4-4 x+x^2}-x} \left (-2 e^5+e^{9-4 x+x^2} (-8+4 x)\right )}{4 e^{2 e^{4-4 x+x^2}-2 x}+e^{5+e^{4-4 x+x^2}-x} (44+16 x)+e^{10} \left (121+88 x+16 x^2\right )} \, dx=2 e^{5} \left (-4 \left (\int \frac {e^{\frac {e^{x^{2}} e^{4}+e^{4 x} x^{2}}{e^{4 x}}}}{4 e^{\frac {2 e^{x^{2}} e^{4}+3 e^{4 x} x}{e^{4 x}}}+16 e^{\frac {e^{x^{2}} e^{4}+4 e^{4 x} x}{e^{4 x}}} e^{5} x +44 e^{\frac {e^{x^{2}} e^{4}+4 e^{4 x} x}{e^{4 x}}} e^{5}+16 e^{5 x} e^{10} x^{2}+88 e^{5 x} e^{10} x +121 e^{5 x} e^{10}}d x \right ) e^{4}-\left (\int \frac {e^{\frac {e^{x^{2}} e^{4}+e^{4 x} x}{e^{4 x}}}}{4 e^{\frac {2 e^{x^{2}} e^{4}}{e^{4 x}}}+16 e^{\frac {e^{x^{2}} e^{4}+e^{4 x} x}{e^{4 x}}} e^{5} x +44 e^{\frac {e^{x^{2}} e^{4}+e^{4 x} x}{e^{4 x}}} e^{5}+16 e^{2 x} e^{10} x^{2}+88 e^{2 x} e^{10} x +121 e^{2 x} e^{10}}d x \right )+2 \left (\int \frac {e^{2 x}}{4 e^{\frac {2 e^{x^{2}} e^{4}}{e^{4 x}}}+16 e^{\frac {e^{x^{2}} e^{4}+e^{4 x} x}{e^{4 x}}} e^{5} x +44 e^{\frac {e^{x^{2}} e^{4}+e^{4 x} x}{e^{4 x}}} e^{5}+16 e^{2 x} e^{10} x^{2}+88 e^{2 x} e^{10} x +121 e^{2 x} e^{10}}d x \right ) e^{5}+2 \left (\int \frac {e^{\frac {e^{x^{2}} e^{4}+e^{4 x} x^{2}}{e^{4 x}}} x}{4 e^{\frac {2 e^{x^{2}} e^{4}+3 e^{4 x} x}{e^{4 x}}}+16 e^{\frac {e^{x^{2}} e^{4}+4 e^{4 x} x}{e^{4 x}}} e^{5} x +44 e^{\frac {e^{x^{2}} e^{4}+4 e^{4 x} x}{e^{4 x}}} e^{5}+16 e^{5 x} e^{10} x^{2}+88 e^{5 x} e^{10} x +121 e^{5 x} e^{10}}d x \right ) e^{4}\right ) \] Input:

int((((4*x-8)*exp(5)*exp(x^2-4*x+4)-2*exp(5))*exp(exp(x^2-4*x+4)-x)+4*exp( 
5)^2)/(4*exp(exp(x^2-4*x+4)-x)^2+(16*x+44)*exp(5)*exp(exp(x^2-4*x+4)-x)+(1 
6*x^2+88*x+121)*exp(5)^2),x)
 

Output:

2*e**5*( - 4*int(e**((e**(x**2)*e**4 + e**(4*x)*x**2)/e**(4*x))/(4*e**((2* 
e**(x**2)*e**4 + 3*e**(4*x)*x)/e**(4*x)) + 16*e**((e**(x**2)*e**4 + 4*e**( 
4*x)*x)/e**(4*x))*e**5*x + 44*e**((e**(x**2)*e**4 + 4*e**(4*x)*x)/e**(4*x) 
)*e**5 + 16*e**(5*x)*e**10*x**2 + 88*e**(5*x)*e**10*x + 121*e**(5*x)*e**10 
),x)*e**4 - int(e**((e**(x**2)*e**4 + e**(4*x)*x)/e**(4*x))/(4*e**((2*e**( 
x**2)*e**4)/e**(4*x)) + 16*e**((e**(x**2)*e**4 + e**(4*x)*x)/e**(4*x))*e** 
5*x + 44*e**((e**(x**2)*e**4 + e**(4*x)*x)/e**(4*x))*e**5 + 16*e**(2*x)*e* 
*10*x**2 + 88*e**(2*x)*e**10*x + 121*e**(2*x)*e**10),x) + 2*int(e**(2*x)/( 
4*e**((2*e**(x**2)*e**4)/e**(4*x)) + 16*e**((e**(x**2)*e**4 + e**(4*x)*x)/ 
e**(4*x))*e**5*x + 44*e**((e**(x**2)*e**4 + e**(4*x)*x)/e**(4*x))*e**5 + 1 
6*e**(2*x)*e**10*x**2 + 88*e**(2*x)*e**10*x + 121*e**(2*x)*e**10),x)*e**5 
+ 2*int((e**((e**(x**2)*e**4 + e**(4*x)*x**2)/e**(4*x))*x)/(4*e**((2*e**(x 
**2)*e**4 + 3*e**(4*x)*x)/e**(4*x)) + 16*e**((e**(x**2)*e**4 + 4*e**(4*x)* 
x)/e**(4*x))*e**5*x + 44*e**((e**(x**2)*e**4 + 4*e**(4*x)*x)/e**(4*x))*e** 
5 + 16*e**(5*x)*e**10*x**2 + 88*e**(5*x)*e**10*x + 121*e**(5*x)*e**10),x)* 
e**4)