\(\int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} (8+6 x-3 e x-4 x^2+e^x (48 x+24 x^2-12 e x^2-12 x^3)+e^{2 x} (-16 x-8 x^2+4 e x^2+4 x^3))+e^{9-6 e^x+e^{2 x}} (-80-60 x+30 e x+40 x^2+e^{2 x} (80 x+40 x^2-20 e x^2-20 x^3)+e^x (-240 x-120 x^2+60 e x^2+60 x^3))}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e (-8 x^4-4 x^5+2 x^6)} \, dx\) [1428]

Optimal result

Integrand size = 226, antiderivative size = 28 \[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=\frac {\left (-5+e^{\left (-3+e^x\right )^2}\right )^2}{x^3 \left (-2+e-\frac {4}{x}+x\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=\frac {e^{-12 e^x} \left (-5 e^{6 e^x}+e^{9+e^{2 x}}\right )^2}{x^2 \left (-4+(-2+e) x+x^2\right )} \] Input:

Integrate[(200 + 150*x - 75*E*x - 100*x^2 + E^(18 - 12*E^x + 2*E^(2*x))*(8 
 + 6*x - 3*E*x - 4*x^2 + E^x*(48*x + 24*x^2 - 12*E*x^2 - 12*x^3) + E^(2*x) 
*(-16*x - 8*x^2 + 4*E*x^2 + 4*x^3)) + E^(9 - 6*E^x + E^(2*x))*(-80 - 60*x 
+ 30*E*x + 40*x^2 + E^(2*x)*(80*x + 40*x^2 - 20*E*x^2 - 20*x^3) + E^x*(-24 
0*x - 120*x^2 + 60*E*x^2 + 60*x^3)))/(16*x^3 + 16*x^4 - 4*x^5 + E^2*x^5 - 
4*x^6 + x^7 + E*(-8*x^4 - 4*x^5 + 2*x^6)),x]
 

Output:

(-5*E^(6*E^x) + E^(9 + E^(2*x)))^2/(E^(12*E^x)*x^2*(-4 + (-2 + E)*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.

\(\displaystyle \int \frac {-100 x^2+e^{-12 e^x+2 e^{2 x}+18} \left (-4 x^2+e^x \left (-12 x^3-12 e x^2+24 x^2+48 x\right )+e^{2 x} \left (4 x^3+4 e x^2-8 x^2-16 x\right )-3 e x+6 x+8\right )+e^{-6 e^x+e^{2 x}+9} \left (40 x^2+e^{2 x} \left (-20 x^3-20 e x^2+40 x^2+80 x\right )+e^x \left (60 x^3+60 e x^2-120 x^2-240 x\right )+30 e x-60 x-80\right )-75 e x+150 x+200}{x^7-4 x^6+e^2 x^5-4 x^5+16 x^4+16 x^3+e \left (2 x^6-4 x^5-8 x^4\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-100 x^2+e^{-12 e^x+2 e^{2 x}+18} \left (-4 x^2+e^x \left (-12 x^3-12 e x^2+24 x^2+48 x\right )+e^{2 x} \left (4 x^3+4 e x^2-8 x^2-16 x\right )-3 e x+6 x+8\right )+e^{-6 e^x+e^{2 x}+9} \left (40 x^2+e^{2 x} \left (-20 x^3-20 e x^2+40 x^2+80 x\right )+e^x \left (60 x^3+60 e x^2-120 x^2-240 x\right )+30 e x-60 x-80\right )-75 e x+150 x+200}{x^7-4 x^6+\left (e^2-4\right ) x^5+16 x^4+16 x^3+e \left (2 x^6-4 x^5-8 x^4\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-100 x^2+e^{-12 e^x+2 e^{2 x}+18} \left (-4 x^2+e^x \left (-12 x^3-12 e x^2+24 x^2+48 x\right )+e^{2 x} \left (4 x^3+4 e x^2-8 x^2-16 x\right )-3 e x+6 x+8\right )+e^{-6 e^x+e^{2 x}+9} \left (40 x^2+e^{2 x} \left (-20 x^3-20 e x^2+40 x^2+80 x\right )+e^x \left (60 x^3+60 e x^2-120 x^2-240 x\right )+30 e x-60 x-80\right )+(150-75 e) x+200}{x^7-4 x^6+\left (e^2-4\right ) x^5+16 x^4+16 x^3+e \left (2 x^6-4 x^5-8 x^4\right )}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {-100 x^2+e^{-12 e^x+2 e^{2 x}+18} \left (-4 x^2+e^x \left (-12 x^3-12 e x^2+24 x^2+48 x\right )+e^{2 x} \left (4 x^3+4 e x^2-8 x^2-16 x\right )-3 e x+6 x+8\right )+e^{-6 e^x+e^{2 x}+9} \left (40 x^2+e^{2 x} \left (-20 x^3-20 e x^2+40 x^2+80 x\right )+e^x \left (60 x^3+60 e x^2-120 x^2-240 x\right )+30 e x-60 x-80\right )+(150-75 e) x+200}{x^3 \left (x^4-2 (2-e) x^3-\left (4+4 e-e^2\right ) x^2+8 (2-e) x+16\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \frac {-100 x^2+e^{-12 e^x+2 e^{2 x}+18} \left (-4 x^2+e^x \left (-12 x^3-12 e x^2+24 x^2+48 x\right )+e^{2 x} \left (4 x^3+4 e x^2-8 x^2-16 x\right )-3 e x+6 x+8\right )+e^{-6 e^x+e^{2 x}+9} \left (40 x^2+e^{2 x} \left (-20 x^3-20 e x^2+40 x^2+80 x\right )+e^x \left (60 x^3+60 e x^2-120 x^2-240 x\right )+30 e x-60 x-80\right )+(150-75 e) x+200}{x^3 \left (x^2+e x-2 x-4\right )^2}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-100 x^2+e^{-12 e^x+2 e^{2 x}+18} \left (-4 x^2+e^x \left (-12 x^3-12 e x^2+24 x^2+48 x\right )+e^{2 x} \left (4 x^3+4 e x^2-8 x^2-16 x\right )-3 e x+6 x+8\right )+e^{-6 e^x+e^{2 x}+9} \left (40 x^2+e^{2 x} \left (-20 x^3-20 e x^2+40 x^2+80 x\right )+e^x \left (60 x^3+60 e x^2-120 x^2-240 x\right )+30 e x-60 x-80\right )+(150-75 e) x+200}{x^3 \left (x^2+(e-2) x-4\right )^2}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-100 x^2+e^{-12 e^x+2 e^{2 x}+18} \left (-4 x^2+e^x \left (-12 x^3-12 e x^2+24 x^2+48 x\right )+e^{2 x} \left (4 x^3+4 e x^2-8 x^2-16 x\right )-3 e x+6 x+8\right )+e^{-6 e^x+e^{2 x}+9} \left (40 x^2+e^{2 x} \left (-20 x^3-20 e x^2+40 x^2+80 x\right )+e^x \left (60 x^3+60 e x^2-120 x^2-240 x\right )+30 e x-60 x-80\right )+(150-75 e) x+200}{x^3 \left (-x^2+(2-e) x+4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 e^{2 x-12 e^x+e^{2 x}+9} \left (5 e^{6 e^x}-e^{e^{2 x}+9}\right )}{x^2 \left (-x^2+(2-e) x+4\right )}+\frac {12 e^{x-12 e^x+e^{2 x}+9} \left (e^{e^{2 x}+9}-5 e^{6 e^x}\right )}{x^2 \left (-x^2+(2-e) x+4\right )}+\frac {40 e^{\left (e^x-3\right )^2}}{x \left (-x^2+(2-e) x+4\right )^2}-\frac {4 e^{2 \left (e^x-3\right )^2}}{x \left (-x^2+(2-e) x+4\right )^2}-\frac {100}{x \left (-x^2+(2-e) x+4\right )^2}-\frac {60 \left (1-\frac {e}{2}\right ) e^{\left (e^x-3\right )^2}}{x^2 \left (-x^2+(2-e) x+4\right )^2}+\frac {6 \left (1-\frac {e}{2}\right ) e^{2 \left (e^x-3\right )^2}}{x^2 \left (-x^2+(2-e) x+4\right )^2}+\frac {75 (2-e)}{x^2 \left (-x^2+(2-e) x+4\right )^2}-\frac {80 e^{\left (e^x-3\right )^2}}{x^3 \left (-x^2+(2-e) x+4\right )^2}+\frac {8 e^{2 \left (e^x-3\right )^2}}{x^3 \left (-x^2+(2-e) x+4\right )^2}+\frac {200}{x^3 \left (-x^2+(2-e) x+4\right )^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 e^{2 x-12 e^x+e^{2 x}+9} \left (5 e^{6 e^x}-e^{e^{2 x}+9}\right )}{x^2 \left (-x^2+(2-e) x+4\right )}+\frac {12 e^{x-12 e^x+e^{2 x}+9} \left (e^{e^{2 x}+9}-5 e^{6 e^x}\right )}{x^2 \left (-x^2+(2-e) x+4\right )}+\frac {40 e^{\left (e^x-3\right )^2}}{x \left (-x^2+(2-e) x+4\right )^2}-\frac {4 e^{2 \left (e^x-3\right )^2}}{x \left (-x^2+(2-e) x+4\right )^2}-\frac {100}{x \left (-x^2+(2-e) x+4\right )^2}+\frac {30 (e-2) e^{\left (e^x-3\right )^2}}{x^2 \left (-x^2+(2-e) x+4\right )^2}+\frac {3 (2-e) e^{2 \left (e^x-3\right )^2}}{x^2 \left (-x^2+(2-e) x+4\right )^2}+\frac {75 (2-e)}{x^2 \left (-x^2+(2-e) x+4\right )^2}-\frac {80 e^{\left (e^x-3\right )^2}}{x^3 \left (-x^2+(2-e) x+4\right )^2}+\frac {8 e^{2 \left (e^x-3\right )^2}}{x^3 \left (-x^2+(2-e) x+4\right )^2}+\frac {200}{x^3 \left (-x^2+(2-e) x+4\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {4 e^{2 x-12 e^x+e^{2 x}+9} \left (5 e^{6 e^x}-e^{e^{2 x}+9}\right )}{x^2 \left (-x^2+(2-e) x+4\right )}+\frac {12 e^{x-12 e^x+e^{2 x}+9} \left (e^{e^{2 x}+9}-5 e^{6 e^x}\right )}{x^2 \left (-x^2+(2-e) x+4\right )}+\frac {40 e^{\left (e^x-3\right )^2}}{x \left (-x^2+(2-e) x+4\right )^2}-\frac {4 e^{2 \left (e^x-3\right )^2}}{x \left (-x^2+(2-e) x+4\right )^2}-\frac {100}{x \left (-x^2+(2-e) x+4\right )^2}+\frac {30 (e-2) e^{\left (e^x-3\right )^2}}{x^2 \left (-x^2+(2-e) x+4\right )^2}+\frac {3 (2-e) e^{2 \left (e^x-3\right )^2}}{x^2 \left (-x^2+(2-e) x+4\right )^2}+\frac {75 (2-e)}{x^2 \left (-x^2+(2-e) x+4\right )^2}-\frac {80 e^{\left (e^x-3\right )^2}}{x^3 \left (-x^2+(2-e) x+4\right )^2}+\frac {8 e^{2 \left (e^x-3\right )^2}}{x^3 \left (-x^2+(2-e) x+4\right )^2}+\frac {200}{x^3 \left (-x^2+(2-e) x+4\right )^2}\right )dx\)

Input:

Int[(200 + 150*x - 75*E*x - 100*x^2 + E^(18 - 12*E^x + 2*E^(2*x))*(8 + 6*x 
 - 3*E*x - 4*x^2 + E^x*(48*x + 24*x^2 - 12*E*x^2 - 12*x^3) + E^(2*x)*(-16* 
x - 8*x^2 + 4*E*x^2 + 4*x^3)) + E^(9 - 6*E^x + E^(2*x))*(-80 - 60*x + 30*E 
*x + 40*x^2 + E^(2*x)*(80*x + 40*x^2 - 20*E*x^2 - 20*x^3) + E^x*(-240*x - 
120*x^2 + 60*E*x^2 + 60*x^3)))/(16*x^3 + 16*x^4 - 4*x^5 + E^2*x^5 - 4*x^6 
+ x^7 + E*(-8*x^4 - 4*x^5 + 2*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.63 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68

method	result	size
parallelrisch	\(\frac {25+{\mathrm e}^{2 \,{\mathrm e}^{2 x}-12 \,{\mathrm e}^{x}+18}-10 \,{\mathrm e}^{{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x}+9}}{x^{2} \left (x \,{\mathrm e}+x^{2}-2 x -4\right )}\)	\(47\)
risch	\(\frac {25}{x^{2} \left (x \,{\mathrm e}+x^{2}-2 x -4\right )}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{2 x}-12 \,{\mathrm e}^{x}+18}}{x^{2} \left (x \,{\mathrm e}+x^{2}-2 x -4\right )}-\frac {10 \,{\mathrm e}^{{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x}+9}}{x^{2} \left (x \,{\mathrm e}+x^{2}-2 x -4\right )}\)	\(82\)

Input:

int((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^3+24*x 
^2+48*x)*exp(x)-3*x*exp(1)-4*x^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+((-20*x 
^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x^2-240*x 
)*exp(x)+30*x*exp(1)+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*x*exp(1)- 
100*x^2+150*x+200)/(x^5*exp(1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^6-4*x^ 
5+16*x^4+16*x^3),x,method=_RETURNVERBOSE)
 

Output:

(25+exp(exp(x)^2-6*exp(x)+9)^2-10*exp(exp(x)^2-6*exp(x)+9))/x^2/(x*exp(1)+ 
x^2-2*x-4)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=\frac {e^{\left (2 \, e^{\left (2 \, x\right )} - 12 \, e^{x} + 18\right )} - 10 \, e^{\left (e^{\left (2 \, x\right )} - 6 \, e^{x} + 9\right )} + 25}{x^{4} + x^{3} e - 2 \, x^{3} - 4 \, x^{2}} \] Input:

integrate((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^ 
3+24*x^2+48*x)*exp(x)-3*exp(1)*x-4*x^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+( 
(-20*x^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x^2 
-240*x)*exp(x)+30*exp(1)*x+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*exp 
(1)*x-100*x^2+150*x+200)/(x^5*exp(1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^ 
6-4*x^5+16*x^4+16*x^3),x, algorithm="fricas")
 

Output:

(e^(2*e^(2*x) - 12*e^x + 18) - 10*e^(e^(2*x) - 6*e^x + 9) + 25)/(x^4 + x^3 
*e - 2*x^3 - 4*x^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (24) = 48\).

Time = 0.78 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=\frac {\left (- 10 x^{4} - 10 e x^{3} + 20 x^{3} + 40 x^{2}\right ) e^{e^{2 x} - 6 e^{x} + 9} + \left (x^{4} - 2 x^{3} + e x^{3} - 4 x^{2}\right ) e^{2 e^{2 x} - 12 e^{x} + 18}}{x^{8} - 4 x^{7} + 2 e x^{7} - 4 e x^{6} - 4 x^{6} + x^{6} e^{2} - 8 e x^{5} + 16 x^{5} + 16 x^{4}} + \frac {25}{x^{4} + x^{3} \left (-2 + e\right ) - 4 x^{2}} \] Input:

integrate((((4*x**2*exp(1)+4*x**3-8*x**2-16*x)*exp(x)**2+(-12*x**2*exp(1)- 
12*x**3+24*x**2+48*x)*exp(x)-3*exp(1)*x-4*x**2+6*x+8)*exp(exp(x)**2-6*exp( 
x)+9)**2+((-20*x**2*exp(1)-20*x**3+40*x**2+80*x)*exp(x)**2+(60*x**2*exp(1) 
+60*x**3-120*x**2-240*x)*exp(x)+30*exp(1)*x+40*x**2-60*x-80)*exp(exp(x)**2 
-6*exp(x)+9)-75*exp(1)*x-100*x**2+150*x+200)/(x**5*exp(1)**2+(2*x**6-4*x** 
5-8*x**4)*exp(1)+x**7-4*x**6-4*x**5+16*x**4+16*x**3),x)
 

Output:

((-10*x**4 - 10*E*x**3 + 20*x**3 + 40*x**2)*exp(exp(2*x) - 6*exp(x) + 9) + 
 (x**4 - 2*x**3 + E*x**3 - 4*x**2)*exp(2*exp(2*x) - 12*exp(x) + 18))/(x**8 
 - 4*x**7 + 2*E*x**7 - 4*E*x**6 - 4*x**6 + x**6*exp(2) - 8*E*x**5 + 16*x** 
5 + 16*x**4) + 25/(x**4 + x**3*(-2 + E) - 4*x**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (27) = 54\).

Time = 0.18 (sec) , antiderivative size = 759, normalized size of antiderivative = 27.11 \[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=\text {Too large to display} \] Input:

integrate((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^ 
3+24*x^2+48*x)*exp(x)-3*exp(1)*x-4*x^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+( 
(-20*x^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x^2 
-240*x)*exp(x)+30*exp(1)*x+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*exp 
(1)*x-100*x^2+150*x+200)/(x^5*exp(1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^ 
6-4*x^5+16*x^4+16*x^3),x, algorithm="maxima")
 

Output:

75/64*((e - 2)*log(x^2 + x*(e - 2) - 4) - 2*(e - 2)*log(x) + (e^4 - 8*e^3 
+ 48*e^2 - 128*e + 208)*log((2*x - sqrt(e^2 - 4*e + 20) + e - 2)/(2*x + sq 
rt(e^2 - 4*e + 20) + e - 2))/(e^2 - 4*e + 20)^(3/2) + 8*(x^2*(e^2 - 4*e + 
16) + x*(e^3 - 6*e^2 + 26*e - 36) - 2*e^2 + 8*e - 40)/(x^3*(e^2 - 4*e + 20 
) + x^2*(e^3 - 6*e^2 + 28*e - 40) - 4*x*(e^2 - 4*e + 20)))*e - 25/64*(3*e^ 
2 - 12*e + 20)*log(x^2 + x*(e - 2) - 4) - 75/32*(e - 2)*log(x^2 + x*(e - 2 
) - 4) + 25/32*(3*e^2 - 12*e + 20)*log(x) + 75/16*(e - 2)*log(x) + (e^(2*e 
^(2*x) + 18) - 10*e^(e^(2*x) + 6*e^x + 9))*e^(-12*e^x)/(x^4 + x^3*(e - 2) 
- 4*x^2) - 25/64*(3*e^5 - 30*e^4 + 200*e^3 - 720*e^2 + 1680*e - 1696)*log( 
(2*x - sqrt(e^2 - 4*e + 20) + e - 2)/(2*x + sqrt(e^2 - 4*e + 20) + e - 2)) 
/(e^2 - 4*e + 20)^(3/2) - 75/32*(e^4 - 8*e^3 + 48*e^2 - 128*e + 208)*log(( 
2*x - sqrt(e^2 - 4*e + 20) + e - 2)/(2*x + sqrt(e^2 - 4*e + 20) + e - 2))/ 
(e^2 - 4*e + 20)^(3/2) + 25/8*(e^3 - 6*e^2 + 36*e - 56)*log((2*x - sqrt(e^ 
2 - 4*e + 20) + e - 2)/(2*x + sqrt(e^2 - 4*e + 20) + e - 2))/(e^2 - 4*e + 
20)^(3/2) - 25/8*(x^3*(3*e^3 - 18*e^2 + 80*e - 112) + x^2*(3*e^4 - 24*e^3 
+ 122*e^2 - 296*e + 312) - 6*x*(e^3 - 6*e^2 + 28*e - 40) - 8*e^2 + 32*e - 
160)/(x^4*(e^2 - 4*e + 20) + x^3*(e^3 - 6*e^2 + 28*e - 40) - 4*x^2*(e^2 - 
4*e + 20)) - 75/4*(x^2*(e^2 - 4*e + 16) + x*(e^3 - 6*e^2 + 26*e - 36) - 2* 
e^2 + 8*e - 40)/(x^3*(e^2 - 4*e + 20) + x^2*(e^3 - 6*e^2 + 28*e - 40) - 4* 
x*(e^2 - 4*e + 20)) + 25*(x*(e - 2) + e^2 - 4*e + 12)/(x^2*(e^2 - 4*e +...
 

Giac [F]

\[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=\int { -\frac {100 \, x^{2} + 75 \, x e + {\left (4 \, x^{2} + 3 \, x e - 4 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} + 12 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{x} - 6 \, x - 8\right )} e^{\left (2 \, e^{\left (2 \, x\right )} - 12 \, e^{x} + 18\right )} - 10 \, {\left (4 \, x^{2} + 3 \, x e - 2 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} + 6 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{x} - 6 \, x - 8\right )} e^{\left (e^{\left (2 \, x\right )} - 6 \, e^{x} + 9\right )} - 150 \, x - 200}{x^{7} - 4 \, x^{6} + x^{5} e^{2} - 4 \, x^{5} + 16 \, x^{4} + 16 \, x^{3} + 2 \, {\left (x^{6} - 2 \, x^{5} - 4 \, x^{4}\right )} e} \,d x } \] Input:

integrate((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^ 
3+24*x^2+48*x)*exp(x)-3*exp(1)*x-4*x^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+( 
(-20*x^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x^2 
-240*x)*exp(x)+30*exp(1)*x+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*exp 
(1)*x-100*x^2+150*x+200)/(x^5*exp(1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^ 
6-4*x^5+16*x^4+16*x^3),x, algorithm="giac")
 

Output:

integrate(-(100*x^2 + 75*x*e + (4*x^2 + 3*x*e - 4*(x^3 + x^2*e - 2*x^2 - 4 
*x)*e^(2*x) + 12*(x^3 + x^2*e - 2*x^2 - 4*x)*e^x - 6*x - 8)*e^(2*e^(2*x) - 
 12*e^x + 18) - 10*(4*x^2 + 3*x*e - 2*(x^3 + x^2*e - 2*x^2 - 4*x)*e^(2*x) 
+ 6*(x^3 + x^2*e - 2*x^2 - 4*x)*e^x - 6*x - 8)*e^(e^(2*x) - 6*e^x + 9) - 1 
50*x - 200)/(x^7 - 4*x^6 + x^5*e^2 - 4*x^5 + 16*x^4 + 16*x^3 + 2*(x^6 - 2* 
x^5 - 4*x^4)*e), x)
 

Mupad [B] (verification not implemented)

Time = 4.43 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=-\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^x+18}\,{\left (5\,{\mathrm {e}}^{6\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}-9}-1\right )}^2}{x^2\,\left (2\,x-x\,\mathrm {e}-x^2+4\right )} \] Input:

int((150*x + exp(2*exp(2*x) - 12*exp(x) + 18)*(6*x - 3*x*exp(1) + exp(x)*( 
48*x - 12*x^2*exp(1) + 24*x^2 - 12*x^3) - 4*x^2 - exp(2*x)*(16*x - 4*x^2*e 
xp(1) + 8*x^2 - 4*x^3) + 8) - 75*x*exp(1) - exp(exp(2*x) - 6*exp(x) + 9)*( 
60*x - 30*x*exp(1) + exp(x)*(240*x - 60*x^2*exp(1) + 120*x^2 - 60*x^3) - 4 
0*x^2 - exp(2*x)*(80*x - 20*x^2*exp(1) + 40*x^2 - 20*x^3) + 80) - 100*x^2 
+ 200)/(x^5*exp(2) - exp(1)*(8*x^4 + 4*x^5 - 2*x^6) + 16*x^3 + 16*x^4 - 4* 
x^5 - 4*x^6 + x^7),x)
 

Output:

-(exp(2*exp(2*x) - 12*exp(x) + 18)*(5*exp(6*exp(x) - exp(2*x) - 9) - 1)^2) 
/(x^2*(2*x - x*exp(1) - x^2 + 4))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx=\frac {e^{2 e^{2 x}} e^{18}-10 e^{e^{2 x}+6 e^{x}} e^{9}+25 e^{12 e^{x}}}{e^{12 e^{x}} x^{2} \left (e x +x^{2}-2 x -4\right )} \] Input:

int((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^3+24*x 
^2+48*x)*exp(x)-3*exp(1)*x-4*x^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+((-20*x 
^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x^2-240*x 
)*exp(x)+30*exp(1)*x+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*exp(1)*x- 
100*x^2+150*x+200)/(x^5*exp(1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^6-4*x^ 
5+16*x^4+16*x^3),x)
 

Output:

(e**(2*e**(2*x))*e**18 - 10*e**(e**(2*x) + 6*e**x)*e**9 + 25*e**(12*e**x)) 
/(e**(12*e**x)*x**2*(e*x + x**2 - 2*x - 4))