\(\int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} (4-4 x^2+x^4)+e^{\frac {e^2}{-2+x^2}} (48-48 x^2+4 e^2 x^2+12 x^4)}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} (8-8 x^2+2 x^4)+e^{\frac {e^2}{-2+x^2}} (80-80 x^2+20 x^4)} \, dx\) [1654]

Optimal result

Integrand size = 139, antiderivative size = 32 \[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\frac {x}{5+e^{\frac {e^2}{x \left (-\frac {2}{x}+x\right )}}}+\frac {1+x}{2} \] Output:

1/2*x+1/2+x/(exp(exp(2)/(x-2/x)/x)+5)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\frac {1}{2} \left (x+\frac {2 x}{5+e^{\frac {e^2}{-2+x^2}}}\right ) \] Input:

Integrate[(140 - 140*x^2 + 35*x^4 + E^((2*E^2)/(-2 + x^2))*(4 - 4*x^2 + x^ 
4) + E^(E^2/(-2 + x^2))*(48 - 48*x^2 + 4*E^2*x^2 + 12*x^4))/(200 - 200*x^2 
 + 50*x^4 + E^((2*E^2)/(-2 + x^2))*(8 - 8*x^2 + 2*x^4) + E^(E^2/(-2 + x^2) 
)*(80 - 80*x^2 + 20*x^4)),x]
 

Output:

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

\(\displaystyle \int \frac {35 x^4-140 x^2+e^{\frac {2 e^2}{x^2-2}} \left (x^4-4 x^2+4\right )+e^{\frac {e^2}{x^2-2}} \left (12 x^4+4 e^2 x^2-48 x^2+48\right )+140}{50 x^4-200 x^2+e^{\frac {2 e^2}{x^2-2}} \left (2 x^4-8 x^2+8\right )+e^{\frac {e^2}{x^2-2}} \left (20 x^4-80 x^2+80\right )+200} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {35 x^4-140 x^2+e^{\frac {2 e^2}{x^2-2}} \left (x^4-4 x^2+4\right )+e^{\frac {e^2}{x^2-2}} \left (12 x^4+4 e^2 x^2-48 x^2+48\right )+140}{2 \left (e^{\frac {e^2}{x^2-2}}+5\right )^2 \left (2-x^2\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {35 x^4-140 x^2+e^{-\frac {2 e^2}{2-x^2}} \left (x^4-4 x^2+4\right )+4 e^{-\frac {e^2}{2-x^2}} \left (3 x^4+e^2 x^2-12 x^2+12\right )+140}{\left (5+e^{-\frac {e^2}{2-x^2}}\right )^2 \left (2-x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{2} \int \left (-\frac {4 e^{2+\frac {2 e^2}{x^2-2}} x^2}{5 \left (5+e^{\frac {e^2}{x^2-2}}\right )^2 \left (x^2-2\right )^2}+\frac {2 e^{\frac {e^2}{x^2-2}} \left (-x^4+2 \left (2+e^2\right ) x^2-4\right )}{5 \left (5+e^{\frac {e^2}{x^2-2}}\right ) \left (2-x^2\right )^2}+\frac {7}{5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{5+e^{\frac {e^2}{x^2-2}}}dx+\frac {1}{5} \sqrt {2} \int \frac {e^{2+\frac {2 e^2}{x^2-2}}}{\left (5+e^{\frac {e^2}{x^2-2}}\right )^2 \left (\sqrt {2}-x\right )}dx-\frac {1}{5} \sqrt {2} \int \frac {e^{2+\frac {e^2}{x^2-2}}}{\left (5+e^{\frac {e^2}{x^2-2}}\right ) \left (\sqrt {2}-x\right )}dx+\frac {1}{5} \sqrt {2} \int \frac {e^{2+\frac {2 e^2}{x^2-2}}}{\left (5+e^{\frac {e^2}{x^2-2}}\right )^2 \left (x+\sqrt {2}\right )}dx-\frac {1}{5} \sqrt {2} \int \frac {e^{2+\frac {e^2}{x^2-2}}}{\left (5+e^{\frac {e^2}{x^2-2}}\right ) \left (x+\sqrt {2}\right )}dx-\frac {8}{5} \int \frac {e^{2+\frac {2 e^2}{x^2-2}}}{\left (5+e^{\frac {e^2}{x^2-2}}\right )^2 \left (x^2-2\right )^2}dx+\frac {8}{5} \int \frac {e^{2+\frac {e^2}{x^2-2}}}{\left (5+e^{\frac {e^2}{x^2-2}}\right ) \left (x^2-2\right )^2}dx+x\right )\)

Input:

Int[(140 - 140*x^2 + 35*x^4 + E^((2*E^2)/(-2 + x^2))*(4 - 4*x^2 + x^4) + E 
^(E^2/(-2 + x^2))*(48 - 48*x^2 + 4*E^2*x^2 + 12*x^4))/(200 - 200*x^2 + 50* 
x^4 + E^((2*E^2)/(-2 + x^2))*(8 - 8*x^2 + 2*x^4) + E^(E^2/(-2 + x^2))*(80 
- 80*x^2 + 20*x^4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {x}{2}+\frac {x}{{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}}+5}\)	\(22\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}} x +7 x}{2 \,{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}}+10}\)	\(35\)
norman	\(\frac {-7 x +\frac {7 x^{3}}{2}-{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}} x +\frac {{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}} x^{3}}{2}}{\left ({\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}}+5\right ) \left (x^{2}-2\right )}\)	\(63\)

Input:

int(((x^4-4*x^2+4)*exp(exp(2)/(x^2-2))^2+(4*x^2*exp(2)+12*x^4-48*x^2+48)*e 
xp(exp(2)/(x^2-2))+35*x^4-140*x^2+140)/((2*x^4-8*x^2+8)*exp(exp(2)/(x^2-2) 
)^2+(20*x^4-80*x^2+80)*exp(exp(2)/(x^2-2))+50*x^4-200*x^2+200),x,method=_R 
ETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\frac {x e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 7 \, x}{2 \, {\left (e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 5\right )}} \] Input:

integrate(((x^4-4*x^2+4)*exp(exp(2)/(x^2-2))^2+(4*x^2*exp(2)+12*x^4-48*x^2 
+48)*exp(exp(2)/(x^2-2))+35*x^4-140*x^2+140)/((2*x^4-8*x^2+8)*exp(exp(2)/( 
x^2-2))^2+(20*x^4-80*x^2+80)*exp(exp(2)/(x^2-2))+50*x^4-200*x^2+200),x, al 
gorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.47 \[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\frac {x}{2} + \frac {x}{e^{\frac {e^{2}}{x^{2} - 2}} + 5} \] Input:

integrate(((x**4-4*x**2+4)*exp(exp(2)/(x**2-2))**2+(4*x**2*exp(2)+12*x**4- 
48*x**2+48)*exp(exp(2)/(x**2-2))+35*x**4-140*x**2+140)/((2*x**4-8*x**2+8)* 
exp(exp(2)/(x**2-2))**2+(20*x**4-80*x**2+80)*exp(exp(2)/(x**2-2))+50*x**4- 
200*x**2+200),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\frac {x e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 7 \, x}{2 \, {\left (e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 5\right )}} \] Input:

integrate(((x^4-4*x^2+4)*exp(exp(2)/(x^2-2))^2+(4*x^2*exp(2)+12*x^4-48*x^2 
+48)*exp(exp(2)/(x^2-2))+35*x^4-140*x^2+140)/((2*x^4-8*x^2+8)*exp(exp(2)/( 
x^2-2))^2+(20*x^4-80*x^2+80)*exp(exp(2)/(x^2-2))+50*x^4-200*x^2+200),x, al 
gorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\int { \frac {35 \, x^{4} - 140 \, x^{2} + {\left (x^{4} - 4 \, x^{2} + 4\right )} e^{\left (\frac {2 \, e^{2}}{x^{2} - 2}\right )} + 4 \, {\left (3 \, x^{4} + x^{2} e^{2} - 12 \, x^{2} + 12\right )} e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 140}{2 \, {\left (25 \, x^{4} - 100 \, x^{2} + {\left (x^{4} - 4 \, x^{2} + 4\right )} e^{\left (\frac {2 \, e^{2}}{x^{2} - 2}\right )} + 10 \, {\left (x^{4} - 4 \, x^{2} + 4\right )} e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 100\right )}} \,d x } \] Input:

integrate(((x^4-4*x^2+4)*exp(exp(2)/(x^2-2))^2+(4*x^2*exp(2)+12*x^4-48*x^2 
+48)*exp(exp(2)/(x^2-2))+35*x^4-140*x^2+140)/((2*x^4-8*x^2+8)*exp(exp(2)/( 
x^2-2))^2+(20*x^4-80*x^2+80)*exp(exp(2)/(x^2-2))+50*x^4-200*x^2+200),x, al 
gorithm="giac")
 

Output:

integrate(1/2*(35*x^4 - 140*x^2 + (x^4 - 4*x^2 + 4)*e^(2*e^2/(x^2 - 2)) + 
4*(3*x^4 + x^2*e^2 - 12*x^2 + 12)*e^(e^2/(x^2 - 2)) + 140)/(25*x^4 - 100*x 
^2 + (x^4 - 4*x^2 + 4)*e^(2*e^2/(x^2 - 2)) + 10*(x^4 - 4*x^2 + 4)*e^(e^2/( 
x^2 - 2)) + 100), x)
 

Mupad [B] (verification not implemented)

Time = 3.94 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\frac {x}{2}+\frac {x}{{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{x^2-2}}+5} \] Input:

int((exp(exp(2)/(x^2 - 2))*(4*x^2*exp(2) - 48*x^2 + 12*x^4 + 48) + exp((2* 
exp(2))/(x^2 - 2))*(x^4 - 4*x^2 + 4) - 140*x^2 + 35*x^4 + 140)/(50*x^4 - 2 
00*x^2 + exp((2*exp(2))/(x^2 - 2))*(2*x^4 - 8*x^2 + 8) + exp(exp(2)/(x^2 - 
 2))*(20*x^4 - 80*x^2 + 80) + 200),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx=\frac {x \left (e^{\frac {e^{2}}{x^{2}-2}}+7\right )}{2 e^{\frac {e^{2}}{x^{2}-2}}+10} \] Input:

int(((x^4-4*x^2+4)*exp(exp(2)/(x^2-2))^2+(4*x^2*exp(2)+12*x^4-48*x^2+48)*e 
xp(exp(2)/(x^2-2))+35*x^4-140*x^2+140)/((2*x^4-8*x^2+8)*exp(exp(2)/(x^2-2) 
)^2+(20*x^4-80*x^2+80)*exp(exp(2)/(x^2-2))+50*x^4-200*x^2+200),x)
 

Output:

(x*(e**(e**2/(x**2 - 2)) + 7))/(2*(e**(e**2/(x**2 - 2)) + 5))