\(\int \frac {30+30 e^4+e^{e-x^2} (5+10 x^2)}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 (36+12 x+x^2)+e^{e-x^2} (12 e^6+e^2 (12+2 x))} \, dx\) [2627]

Optimal result

Integrand size = 91, antiderivative size = 25 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=\frac {5 x}{e^2 \left (6+6 e^4+e^{e-x^2}+x\right )} \] Output:

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

Mathematica [A] (verified)

Time = 7.55 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=-\frac {5 \left (e^e+6 e^{x^2}+6 e^{4+x^2}\right )}{e^2 \left (e^e+6 e^{4+x^2}+e^{x^2} (6+x)\right )} \] Input:

Integrate[(30 + 30*E^4 + E^(E - x^2)*(5 + 10*x^2))/(36*E^10 + E^(2 + 2*E - 
 2*x^2) + E^6*(72 + 12*x) + E^2*(36 + 12*x + x^2) + E^(E - x^2)*(12*E^6 + 
E^2*(12 + 2*x))),x]
 

Output:

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

\(\displaystyle \int \frac {e^{e-x^2} \left (10 x^2+5\right )+30 e^4+30}{e^{-2 x^2+2 e+2}+e^2 \left (x^2+12 x+36\right )+e^{e-x^2} \left (e^2 (2 x+12)+12 e^6\right )+e^6 (12 x+72)+36 e^{10}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{2 x^2-2} \left (e^{e-x^2} \left (10 x^2+5\right )+30 \left (1+e^4\right )\right )}{\left (e^{x^2} x+6 \left (1+e^4\right ) e^{x^2}+e^e\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {5 e^{2 x^2-e-2} \left (x+6 e^4+6\right ) \left (-2 x^2-1\right )}{e^{x^2} x+6 \left (1+e^4\right ) e^{x^2}+e^e}+\frac {5 e^{2 x^2-2} x \left (-2 x^2-12 \left (1+e^4\right ) x-1\right )}{\left (e^{x^2} x+6 \left (1+e^4\right ) e^{x^2}+e^e\right )^2}+5 e^{x^2-e-2} \left (2 x^2+1\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 5 \int \frac {e^{2 x^2-e-2} x}{-e^{x^2} x-6 e^{x^2} \left (1+e^4\right )-e^e}dx-5 \int \frac {e^{2 x^2-2} x}{\left (e^{x^2} x+6 e^{x^2} \left (1+e^4\right )+e^e\right )^2}dx-60 \left (1+e^4\right ) \int \frac {e^{2 x^2-2} x^2}{\left (e^{x^2} x+6 e^{x^2} \left (1+e^4\right )+e^e\right )^2}dx-30 \left (1+e^4\right ) \int \frac {e^{2 x^2-e-2}}{e^{x^2} x+6 e^{x^2} \left (1+e^4\right )+e^e}dx-60 \left (1+e^4\right ) \int \frac {e^{2 x^2-e-2} x^2}{e^{x^2} x+6 e^{x^2} \left (1+e^4\right )+e^e}dx+10 \int \frac {e^{2 x^2-e-2} x^3}{-e^{x^2} x-6 e^{x^2} \left (1+e^4\right )-e^e}dx-10 \int \frac {e^{2 x^2-2} x^3}{\left (e^{x^2} x+6 e^{x^2} \left (1+e^4\right )+e^e\right )^2}dx+5 e^{x^2-e-2} x\)

Input:

Int[(30 + 30*E^4 + E^(E - x^2)*(5 + 10*x^2))/(36*E^10 + E^(2 + 2*E - 2*x^2 
) + E^6*(72 + 12*x) + E^2*(36 + 12*x + x^2) + E^(E - x^2)*(12*E^6 + E^2*(1 
2 + 2*x))),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {5 x \,{\mathrm e}^{-2}}{{\mathrm e}^{{\mathrm e}-x^{2}}+6+x +6 \,{\mathrm e}^{4}}\)	\(24\)
norman	\(\frac {5 x \,{\mathrm e}^{-2}}{{\mathrm e}^{{\mathrm e}-x^{2}}+6+x +6 \,{\mathrm e}^{4}}\)	\(26\)
parallelrisch	\(\frac {5 x \,{\mathrm e}^{-2}}{{\mathrm e}^{{\mathrm e}-x^{2}}+6+x +6 \,{\mathrm e}^{4}}\)	\(26\)

Input:

int(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2)^2+(1 
2*exp(2)*exp(4)+(2*x+12)*exp(2))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+(12*x+ 
72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=\frac {5 \, x}{{\left (x + 6\right )} e^{2} + 6 \, e^{6} + e^{\left (-x^{2} + e + 2\right )}} \] Input:

integrate(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2 
)^2+(12*exp(2)*exp(4)+(2*x+12)*exp(2))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+ 
(12*x+72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x, algorithm="fricas")
 

Output:

5*x/((x + 6)*e^2 + 6*e^6 + e^(-x^2 + e + 2))
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=\frac {5 x}{x e^{2} + e^{2} e^{e - x^{2}} + 6 e^{2} + 6 e^{6}} \] Input:

integrate(((10*x**2+5)*exp(exp(1)-x**2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x 
**2)**2+(12*exp(2)*exp(4)+(2*x+12)*exp(2))*exp(exp(1)-x**2)+36*exp(2)*exp( 
4)**2+(12*x+72)*exp(2)*exp(4)+(x**2+12*x+36)*exp(2)),x)
 

Output:

5*x/(x*exp(2) + exp(2)*exp(E - x**2) + 6*exp(2) + 6*exp(6))
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=-\frac {5 \, {\left (6 \, {\left (e^{4} + 1\right )} e^{\left (x^{2}\right )} + e^{e}\right )}}{{\left (x e^{2} + 6 \, e^{6} + 6 \, e^{2}\right )} e^{\left (x^{2}\right )} + e^{\left (e + 2\right )}} \] Input:

integrate(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2 
)^2+(12*exp(2)*exp(4)+(2*x+12)*exp(2))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+ 
(12*x+72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x, algorithm="maxima")
 

Output:

-5*(6*(e^4 + 1)*e^(x^2) + e^e)/((x*e^2 + 6*e^6 + 6*e^2)*e^(x^2) + e^(e + 2 
))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (23) = 46\).

Time = 0.13 (sec) , antiderivative size = 501, normalized size of antiderivative = 20.04 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=-\frac {5 \, {\left (12 \, x^{3} e^{\left (x^{2} + 6\right )} + 12 \, x^{3} e^{\left (x^{2} + 2\right )} + 2 \, x^{3} e^{\left (e + 2\right )} + 144 \, x^{2} e^{\left (x^{2} + 10\right )} + 288 \, x^{2} e^{\left (x^{2} + 6\right )} + 144 \, x^{2} e^{\left (x^{2} + 2\right )} + 2 \, x^{2} e^{\left (-x^{2} + 2 \, e + 2\right )} + 36 \, x^{2} e^{\left (e + 6\right )} + 36 \, x^{2} e^{\left (e + 2\right )} + 432 \, x e^{\left (x^{2} + 14\right )} + 1296 \, x e^{\left (x^{2} + 10\right )} + 1302 \, x e^{\left (x^{2} + 6\right )} + 438 \, x e^{\left (x^{2} + 2\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 6\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 2\right )} + 144 \, x e^{\left (e + 10\right )} + 288 \, x e^{\left (e + 6\right )} + 145 \, x e^{\left (e + 2\right )} + 36 \, e^{\left (x^{2} + 10\right )} + 72 \, e^{\left (x^{2} + 6\right )} + 36 \, e^{\left (x^{2} + 2\right )} + e^{\left (-x^{2} + 2 \, e + 2\right )} + 12 \, e^{\left (e + 6\right )} + 12 \, e^{\left (e + 2\right )}\right )}}{2 \, x^{4} e^{\left (x^{2} + 4\right )} + 36 \, x^{3} e^{\left (x^{2} + 8\right )} + 36 \, x^{3} e^{\left (x^{2} + 4\right )} + 4 \, x^{3} e^{\left (e + 4\right )} + 216 \, x^{2} e^{\left (x^{2} + 12\right )} + 432 \, x^{2} e^{\left (x^{2} + 8\right )} + 217 \, x^{2} e^{\left (x^{2} + 4\right )} + 2 \, x^{2} e^{\left (-x^{2} + 2 \, e + 4\right )} + 48 \, x^{2} e^{\left (e + 8\right )} + 48 \, x^{2} e^{\left (e + 4\right )} + 432 \, x e^{\left (x^{2} + 16\right )} + 1296 \, x e^{\left (x^{2} + 12\right )} + 1308 \, x e^{\left (x^{2} + 8\right )} + 444 \, x e^{\left (x^{2} + 4\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 8\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 4\right )} + 144 \, x e^{\left (e + 12\right )} + 288 \, x e^{\left (e + 8\right )} + 146 \, x e^{\left (e + 4\right )} + 36 \, e^{\left (x^{2} + 12\right )} + 72 \, e^{\left (x^{2} + 8\right )} + 36 \, e^{\left (x^{2} + 4\right )} + e^{\left (-x^{2} + 2 \, e + 4\right )} + 12 \, e^{\left (e + 8\right )} + 12 \, e^{\left (e + 4\right )}} \] Input:

integrate(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2 
)^2+(12*exp(2)*exp(4)+(2*x+12)*exp(2))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+ 
(12*x+72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x, algorithm="giac")
 

Output:

-5*(12*x^3*e^(x^2 + 6) + 12*x^3*e^(x^2 + 2) + 2*x^3*e^(e + 2) + 144*x^2*e^ 
(x^2 + 10) + 288*x^2*e^(x^2 + 6) + 144*x^2*e^(x^2 + 2) + 2*x^2*e^(-x^2 + 2 
*e + 2) + 36*x^2*e^(e + 6) + 36*x^2*e^(e + 2) + 432*x*e^(x^2 + 14) + 1296* 
x*e^(x^2 + 10) + 1302*x*e^(x^2 + 6) + 438*x*e^(x^2 + 2) + 12*x*e^(-x^2 + 2 
*e + 6) + 12*x*e^(-x^2 + 2*e + 2) + 144*x*e^(e + 10) + 288*x*e^(e + 6) + 1 
45*x*e^(e + 2) + 36*e^(x^2 + 10) + 72*e^(x^2 + 6) + 36*e^(x^2 + 2) + e^(-x 
^2 + 2*e + 2) + 12*e^(e + 6) + 12*e^(e + 2))/(2*x^4*e^(x^2 + 4) + 36*x^3*e 
^(x^2 + 8) + 36*x^3*e^(x^2 + 4) + 4*x^3*e^(e + 4) + 216*x^2*e^(x^2 + 12) + 
 432*x^2*e^(x^2 + 8) + 217*x^2*e^(x^2 + 4) + 2*x^2*e^(-x^2 + 2*e + 4) + 48 
*x^2*e^(e + 8) + 48*x^2*e^(e + 4) + 432*x*e^(x^2 + 16) + 1296*x*e^(x^2 + 1 
2) + 1308*x*e^(x^2 + 8) + 444*x*e^(x^2 + 4) + 12*x*e^(-x^2 + 2*e + 8) + 12 
*x*e^(-x^2 + 2*e + 4) + 144*x*e^(e + 12) + 288*x*e^(e + 8) + 146*x*e^(e + 
4) + 36*e^(x^2 + 12) + 72*e^(x^2 + 8) + 36*e^(x^2 + 4) + e^(-x^2 + 2*e + 4 
) + 12*e^(e + 8) + 12*e^(e + 4))
 

Mupad [B] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=\frac {x\,\left (30\,{\mathrm {e}}^4+30\right )}{6\,\left ({\mathrm {e}}^4+1\right )\,\left ({\mathrm {e}}^{-x^2+\mathrm {e}+2}+6\,{\mathrm {e}}^2+6\,{\mathrm {e}}^6+x\,{\mathrm {e}}^2\right )} \] Input:

int((30*exp(4) + exp(exp(1) - x^2)*(10*x^2 + 5) + 30)/(36*exp(10) + exp(2) 
*exp(2*exp(1) - 2*x^2) + exp(2)*(12*x + x^2 + 36) + exp(exp(1) - x^2)*(12* 
exp(6) + exp(2)*(2*x + 12)) + exp(6)*(12*x + 72)),x)
 

Output:

(x*(30*exp(4) + 30))/(6*(exp(4) + 1)*(exp(exp(1) - x^2 + 2) + 6*exp(2) + 6 
*exp(6) + x*exp(2)))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx=\frac {5 e^{x^{2}} x}{e^{2} \left (6 e^{x^{2}} e^{4}+e^{x^{2}} x +6 e^{x^{2}}+e^{e}\right )} \] Input:

int(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2)^2+(1 
2*exp(2)*exp(4)+(2*x+12)*exp(2))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+(12*x+ 
72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x)
 

Output:

(5*e**(x**2)*x)/(e**2*(6*e**(x**2)*e**4 + e**(x**2)*x + 6*e**(x**2) + e**e 
))