\(\int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} (-2 e+2 e^x x)}{e^{1+\frac {3 (2 e-e^x)}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 (2 e-e^x)}{e}} (-3 e^{1+x} x^3-3 e x^4)+e^{\frac {2 e-e^x}{e}} (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5)} \, dx\) [2727]

Optimal result

Integrand size = 187, antiderivative size = 26 \[ \int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx=\frac {1}{\left (e^{2-e^{-1+x}}-e^x-x\right )^2 x^2} \] Output:

1/x^2/(exp(2-exp(x)/exp(1))-exp(x)-x)^2
 

Mathematica [A] (verified)

Time = 2.97 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx=\frac {e^{2 e^{-1+x}}}{x^2 \left (-e^2+e^{e^{-1+x}+x}+e^{e^{-1+x}} x\right )^2} \] Input:

Integrate[(4*E*x + E^(1 + x)*(2 + 2*x) + E^((2*E - E^x)/E)*(-2*E + 2*E^x*x 
))/(E^(1 + (3*(2*E - E^x))/E)*x^3 - E^(1 + 3*x)*x^3 - 3*E^(1 + 2*x)*x^4 - 
3*E^(1 + x)*x^5 - E*x^6 + E^((2*(2*E - E^x))/E)*(-3*E^(1 + x)*x^3 - 3*E*x^ 
4) + E^((2*E - E^x)/E)*(3*E^(1 + 2*x)*x^3 + 6*E^(1 + x)*x^4 + 3*E*x^5)),x]
 

Output:

E^(2*E^(-1 + x))/(x^2*(-E^2 + E^(E^(-1 + x) + x) + E^E^(-1 + 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.

\(\displaystyle \int \frac {4 e x+e^{x+1} (2 x+2)+e^{\frac {2 e-e^x}{e}} \left (2 e^x x-2 e\right )}{-e x^6-3 e^{x+1} x^5-3 e^{2 x+1} x^4+e^{\frac {3 \left (2 e-e^x\right )}{e}+1} x^3-e^{3 x+1} x^3+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e x^4-3 e^{x+1} x^3\right )+e^{\frac {2 e-e^x}{e}} \left (3 e x^5+6 e^{x+1} x^4+3 e^{2 x+1} x^3\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(4*E*x + E^(1 + x)*(2 + 2*x) + E^((2*E - E^x)/E)*(-2*E + 2*E^x*x))/(E^ 
(1 + (3*(2*E - E^x))/E)*x^3 - E^(1 + 3*x)*x^3 - 3*E^(1 + 2*x)*x^4 - 3*E^(1 
 + x)*x^5 - E*x^6 + E^((2*(2*E - E^x))/E)*(-3*E^(1 + x)*x^3 - 3*E*x^4) + E 
^((2*E - E^x)/E)*(3*E^(1 + 2*x)*x^3 + 6*E^(1 + x)*x^4 + 3*E*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {1}{x^{2} \left ({\mathrm e}^{x}-{\mathrm e}^{\left (-{\mathrm e}^{x}+2 \,{\mathrm e}\right ) {\mathrm e}^{-1}}+x \right )^{2}}\)	\(26\)
parallelrisch	\(\frac {1}{x^{2} \left (x^{2}+2 \,{\mathrm e}^{x} x -2 x \,{\mathrm e}^{\left (-{\mathrm e}^{x}+2 \,{\mathrm e}\right ) {\mathrm e}^{-1}}+{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} {\mathrm e}^{\left (-{\mathrm e}^{x}+2 \,{\mathrm e}\right ) {\mathrm e}^{-1}}+{\mathrm e}^{2 \left (-{\mathrm e}^{x}+2 \,{\mathrm e}\right ) {\mathrm e}^{-1}}\right )}\)	\(74\)

Input:

int(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2+2*x)*exp(1)*e 
xp(x)+4*x*exp(1))/(x^3*exp(1)*exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x^3*exp 
(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*exp(1)*ex 
p(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))-x^ 
3*exp(1)*exp(x)^3-3*x^4*exp(1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp(1)),x, 
method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (21) = 42\).

Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.31 \[ \int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx=\frac {e^{2}}{x^{4} e^{2} + 2 \, x^{3} e^{\left (x + 2\right )} + x^{2} e^{\left (\frac {2}{3} \, {\left (7 \, e^{2} - 3 \, e^{\left (x + 1\right )}\right )} e^{\left (-2\right )} + \frac {4}{3}\right )} + x^{2} e^{\left (2 \, x + 2\right )} - 2 \, {\left (x^{3} e^{\frac {5}{3}} + x^{2} e^{\left (x + \frac {5}{3}\right )}\right )} e^{\left (\frac {1}{3} \, {\left (7 \, e^{2} - 3 \, e^{\left (x + 1\right )}\right )} e^{\left (-2\right )}\right )}} \] Input:

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2+2*x)*ex 
p(1)*exp(x)+4*exp(1)*x)/(x^3*exp(1)*exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x 
^3*exp(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*exp 
(1)*exp(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp( 
1))-x^3*exp(1)*exp(x)^3-3*x^4*exp(1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp( 
1)),x, algorithm="fricas")
 

Output:

e^2/(x^4*e^2 + 2*x^3*e^(x + 2) + x^2*e^(2/3*(7*e^2 - 3*e^(x + 1))*e^(-2) + 
 4/3) + x^2*e^(2*x + 2) - 2*(x^3*e^(5/3) + x^2*e^(x + 5/3))*e^(1/3*(7*e^2 
- 3*e^(x + 1))*e^(-2)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).

Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx=\frac {1}{x^{4} + 2 x^{3} e^{x} + x^{2} e^{2 x} + x^{2} e^{\frac {2 \left (- e^{x} + 2 e\right )}{e}} + \left (- 2 x^{3} - 2 x^{2} e^{x}\right ) e^{\frac {- e^{x} + 2 e}{e}}} \] Input:

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2+2*x)*ex 
p(1)*exp(x)+4*exp(1)*x)/(x**3*exp(1)*exp((-exp(x)+2*exp(1))/exp(1))**3+(-3 
*x**3*exp(1)*exp(x)-3*x**4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))**2+(3*x* 
*3*exp(1)*exp(x)**2+6*x**4*exp(1)*exp(x)+3*x**5*exp(1))*exp((-exp(x)+2*exp 
(1))/exp(1))-x**3*exp(1)*exp(x)**3-3*x**4*exp(1)*exp(x)**2-3*x**5*exp(1)*e 
xp(x)-x**6*exp(1)),x)
 

Output:

1/(x**4 + 2*x**3*exp(x) + x**2*exp(2*x) + x**2*exp(2*(-exp(x) + 2*E)*exp(- 
1)) + (-2*x**3 - 2*x**2*exp(x))*exp((-exp(x) + 2*E)*exp(-1)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).

Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx=\frac {e^{\left (2 \, e^{\left (x - 1\right )}\right )}}{x^{2} e^{4} + {\left (x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{\left (x - 1\right )}\right )} - 2 \, {\left (x^{3} e^{2} + x^{2} e^{\left (x + 2\right )}\right )} e^{\left (e^{\left (x - 1\right )}\right )}} \] Input:

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2+2*x)*ex 
p(1)*exp(x)+4*exp(1)*x)/(x^3*exp(1)*exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x 
^3*exp(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*exp 
(1)*exp(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp( 
1))-x^3*exp(1)*exp(x)^3-3*x^4*exp(1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp( 
1)),x, algorithm="maxima")
 

Output:

e^(2*e^(x - 1))/(x^2*e^4 + (x^4 + 2*x^3*e^x + x^2*e^(2*x))*e^(2*e^(x - 1)) 
 - 2*(x^3*e^2 + x^2*e^(x + 2))*e^(e^(x - 1)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123800 vs. \(2 (21) = 42\).

Time = 73.07 (sec) , antiderivative size = 123800, normalized size of antiderivative = 4761.54 \[ \int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx=\text {Too large to display} \] Input:

integrate(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2+2*x)*ex 
p(1)*exp(x)+4*exp(1)*x)/(x^3*exp(1)*exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x 
^3*exp(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*exp 
(1)*exp(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp( 
1))-x^3*exp(1)*exp(x)^3-3*x^4*exp(1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp( 
1)),x, algorithm="giac")
 

Output:

((x + 1)^13*e^(12*x + 2*e^(x - 1) + 13) + 11*(x + 1)^12*e^(13*x + 2*e^(x - 
 1) + 13) + 9*(x + 1)^12*e^(12*x + 2*e^(x - 1) + 14) - 13*(x + 1)^12*e^(12 
*x + 2*e^(x - 1) + 13) - 2*(x + 1)^12*e^(12*x + e^(x - 1) + 15) + 9*(x + 1 
)^12*e^(11*x + 2*e^(x - 1) + 14) + 55*(x + 1)^11*e^(14*x + 2*e^(x - 1) + 1 
3) + 90*(x + 1)^11*e^(13*x + 2*e^(x - 1) + 14) - 132*(x + 1)^11*e^(13*x + 
2*e^(x - 1) + 13) - 20*(x + 1)^11*e^(13*x + e^(x - 1) + 15) + 36*(x + 1)^1 
1*e^(12*x + 2*e^(x - 1) + 15) - 18*(x + 1)^11*e^(12*x + 2*e^(x - 1) + 14) 
+ 78*(x + 1)^11*e^(12*x + 2*e^(x - 1) + 13) - 18*(x + 1)^11*e^(12*x + e^(x 
 - 1) + 16) + 24*(x + 1)^11*e^(12*x + e^(x - 1) + 15) + (x + 1)^11*e^(12*x 
 + 17) + 72*(x + 1)^11*e^(11*x + 2*e^(x - 1) + 15) - 108*(x + 1)^11*e^(11* 
x + 2*e^(x - 1) + 14) - 18*(x + 1)^11*e^(11*x + e^(x - 1) + 16) + 36*(x + 
1)^11*e^(10*x + 2*e^(x - 1) + 15) + 165*(x + 1)^10*e^(15*x + 2*e^(x - 1) + 
 13) + 405*(x + 1)^10*e^(14*x + 2*e^(x - 1) + 14) - 605*(x + 1)^10*e^(14*x 
 + 2*e^(x - 1) + 13) - 90*(x + 1)^10*e^(14*x + e^(x - 1) + 15) + 324*(x + 
1)^10*e^(13*x + 2*e^(x - 1) + 15) - 585*(x + 1)^10*e^(13*x + 2*e^(x - 1) + 
 14) + 726*(x + 1)^10*e^(13*x + 2*e^(x - 1) + 13) - 162*(x + 1)^10*e^(13*x 
 + e^(x - 1) + 16) + 220*(x + 1)^10*e^(13*x + e^(x - 1) + 15) + 9*(x + 1)^ 
10*e^(13*x + 17) + 84*(x + 1)^10*e^(12*x + 2*e^(x - 1) + 16) + 252*(x + 1) 
^10*e^(12*x + 2*e^(x - 1) + 15) - 396*(x + 1)^10*e^(12*x + 2*e^(x - 1) + 1 
4) - 286*(x + 1)^10*e^(12*x + 2*e^(x - 1) + 13) - 72*(x + 1)^10*e^(12*x...
 

Mupad [F(-1)]

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

int(-(4*x*exp(1) - exp(exp(-1)*(2*exp(1) - exp(x)))*(2*exp(1) - 2*x*exp(x) 
) + exp(1)*exp(x)*(2*x + 2))/(exp(2*exp(-1)*(2*exp(1) - exp(x)))*(3*x^4*ex 
p(1) + 3*x^3*exp(1)*exp(x)) - exp(exp(-1)*(2*exp(1) - exp(x)))*(3*x^5*exp( 
1) + 6*x^4*exp(1)*exp(x) + 3*x^3*exp(2*x)*exp(1)) + x^6*exp(1) - x^3*exp(1 
)*exp(3*exp(-1)*(2*exp(1) - exp(x))) + 3*x^5*exp(1)*exp(x) + x^3*exp(3*x)* 
exp(1) + 3*x^4*exp(2*x)*exp(1)),x)
 

Output:

int(-(4*x*exp(1) - exp(exp(-1)*(2*exp(1) - exp(x)))*(2*exp(1) - 2*x*exp(x) 
) + exp(1)*exp(x)*(2*x + 2))/(exp(2*exp(-1)*(2*exp(1) - exp(x)))*(3*x^4*ex 
p(1) + 3*x^3*exp(1)*exp(x)) - exp(exp(-1)*(2*exp(1) - exp(x)))*(3*x^5*exp( 
1) + 6*x^4*exp(1)*exp(x) + 3*x^3*exp(2*x)*exp(1)) + x^6*exp(1) - x^3*exp(1 
)*exp(3*exp(-1)*(2*exp(1) - exp(x))) + 3*x^5*exp(1)*exp(x) + x^3*exp(3*x)* 
exp(1) + 3*x^4*exp(2*x)*exp(1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.88 \[ \int \frac {4 e x+e^{1+x} (2+2 x)+e^{\frac {2 e-e^x}{e}} \left (-2 e+2 e^x x\right )}{e^{1+\frac {3 \left (2 e-e^x\right )}{e}} x^3-e^{1+3 x} x^3-3 e^{1+2 x} x^4-3 e^{1+x} x^5-e x^6+e^{\frac {2 \left (2 e-e^x\right )}{e}} \left (-3 e^{1+x} x^3-3 e x^4\right )+e^{\frac {2 e-e^x}{e}} \left (3 e^{1+2 x} x^3+6 e^{1+x} x^4+3 e x^5\right )} \, dx=\frac {e^{\frac {2 e^{x}}{e}}}{x^{2} \left (e^{\frac {2 e^{x}+2 e x}{e}}+2 e^{\frac {2 e^{x}+e x}{e}} x +e^{\frac {2 e^{x}}{e}} x^{2}-2 e^{\frac {e^{x}+e x}{e}} e^{2}-2 e^{\frac {e^{x}}{e}} e^{2} x +e^{4}\right )} \] Input:

int(((2*exp(x)*x-2*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))+(2+2*x)*exp(1)*e 
xp(x)+4*exp(1)*x)/(x^3*exp(1)*exp((-exp(x)+2*exp(1))/exp(1))^3+(-3*x^3*exp 
(1)*exp(x)-3*x^4*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))^2+(3*x^3*exp(1)*ex 
p(x)^2+6*x^4*exp(1)*exp(x)+3*x^5*exp(1))*exp((-exp(x)+2*exp(1))/exp(1))-x^ 
3*exp(1)*exp(x)^3-3*x^4*exp(1)*exp(x)^2-3*x^5*exp(1)*exp(x)-x^6*exp(1)),x)
 

Output:

e**((2*e**x)/e)/(x**2*(e**((2*e**x + 2*e*x)/e) + 2*e**((2*e**x + e*x)/e)*x 
 + e**((2*e**x)/e)*x**2 - 2*e**((e**x + e*x)/e)*e**2 - 2*e**(e**x/e)*e**2* 
x + e**4))