\(\int \frac {e^x (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7))}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8)+e^{\frac {2 x^2}{1+3 x+16 x^3}} (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9)} \, dx\) [728]

Optimal result

Integrand size = 231, antiderivative size = 32 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {e^x}{\left (e^{\frac {2 x}{3+\frac {1}{x}+16 x^2}}+\frac {x}{4}\right ) x} \] Output:

exp(x)/(exp(2*x/(3+1/x+16*x^2))+1/4*x)/x
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \] Input:

Integrate[(E^x*(-8*x - 44*x^2 - 48*x^3 - 220*x^4 - 640*x^5 + 384*x^6 - 204 
8*x^7 + 1024*x^8 + E^((2*x^2)/(1 + 3*x + 16*x^3))*(-16 - 80*x - 112*x^2 - 
464*x^3 - 1024*x^4 + 2048*x^5 - 4096*x^6 + 4096*x^7)))/(x^4 + 6*x^5 + 9*x^ 
6 + 32*x^7 + 96*x^8 + 256*x^10 + E^((4*x^2)/(1 + 3*x + 16*x^3))*(16*x^2 + 
96*x^3 + 144*x^4 + 512*x^5 + 1536*x^6 + 4096*x^8) + E^((2*x^2)/(1 + 3*x + 
16*x^3))*(8*x^3 + 48*x^4 + 72*x^5 + 256*x^6 + 768*x^7 + 2048*x^9)),x]
 

Output:

(4*E^x)/(x*(4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + 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.

\(\displaystyle \int \frac {e^x \left (1024 x^8-2048 x^7+384 x^6-640 x^5-220 x^4-48 x^3-44 x^2+e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (4096 x^7-4096 x^6+2048 x^5-1024 x^4-464 x^3-112 x^2-80 x-16\right )-8 x\right )}{256 x^{10}+96 x^8+32 x^7+9 x^6+6 x^5+x^4+e^{\frac {4 x^2}{16 x^3+3 x+1}} \left (4096 x^8+1536 x^6+512 x^5+144 x^4+96 x^3+16 x^2\right )+e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (2048 x^9+768 x^7+256 x^6+72 x^5+48 x^4+8 x^3\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 e^x \left ((x-2) x \left (16 x^3+3 x+1\right )^2+4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (256 x^7-256 x^6+128 x^5-64 x^4-29 x^3-7 x^2-5 x-1\right )\right )}{x^2 \left (4 e^{\frac {2 x^2}{16 x^3+3 x+1}}+x\right )^2 \left (16 x^3+3 x+1\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int -\frac {e^x \left ((2-x) x \left (16 x^3+3 x+1\right )^2+4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (-256 x^7+256 x^6-128 x^5+64 x^4+29 x^3+7 x^2+5 x+1\right )\right )}{x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 \left (16 x^3+3 x+1\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \int \frac {e^x \left ((2-x) x \left (16 x^3+3 x+1\right )^2+4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (-256 x^7+256 x^6-128 x^5+64 x^4+29 x^3+7 x^2+5 x+1\right )\right )}{x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 \left (16 x^3+3 x+1\right )^2}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle -4 \int \left (\frac {4 e^x \left ((2-x) x \left (16 x^3+3 x+1\right )^2+4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (-256 x^7+256 x^6-128 x^5+64 x^4+29 x^3+7 x^2+5 x+1\right )\right )}{9 x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 (4 x+1)}+\frac {e^x (1-4 x) \left ((2-x) x \left (16 x^3+3 x+1\right )^2+4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (-256 x^7+256 x^6-128 x^5+64 x^4+29 x^3+7 x^2+5 x+1\right )\right )}{9 x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 \left (4 x^2-x+1\right )}+\frac {4 e^x \left ((2-x) x \left (16 x^3+3 x+1\right )^2+4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (-256 x^7+256 x^6-128 x^5+64 x^4+29 x^3+7 x^2+5 x+1\right )\right )}{9 x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 (4 x+1)^2}-\frac {e^x \left ((2-x) x \left (16 x^3+3 x+1\right )^2+4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (-256 x^7+256 x^6-128 x^5+64 x^4+29 x^3+7 x^2+5 x+1\right )\right )}{3 x \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 \left (4 x^2-x+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -4 \int \frac {e^x \left (-\left ((x-2) x \left (16 x^3+3 x+1\right )^2\right )-4 e^{\frac {2 x^2}{16 x^3+3 x+1}} \left (256 x^7-256 x^6+128 x^5-64 x^4-29 x^3-7 x^2-5 x-1\right )\right )}{x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 (4 x+1)^2 \left (4 x^2-x+1\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (\frac {e^x \left (256 x^6+32 x^5+96 x^4+26 x^3+5 x^2+6 x+1\right )}{x \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 (4 x+1)^2 \left (4 x^2-x+1\right )^2}-\frac {e^x \left (256 x^7-256 x^6+128 x^5-64 x^4-29 x^3-7 x^2-5 x-1\right )}{x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right ) \left (16 x^3+3 x+1\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle -4 \int \left (\frac {e^x \left (256 x^6+32 x^5+96 x^4+26 x^3+5 x^2+6 x+1\right )}{x \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right )^2 (4 x+1)^2 \left (4 x^2-x+1\right )^2}-\frac {e^x \left (256 x^7-256 x^6+128 x^5-64 x^4-29 x^3-7 x^2-5 x-1\right )}{x^2 \left (x+4 e^{\frac {2 x^2}{16 x^3+3 x+1}}\right ) \left (16 x^3+3 x+1\right )^2}\right )dx\)

Input:

Int[(E^x*(-8*x - 44*x^2 - 48*x^3 - 220*x^4 - 640*x^5 + 384*x^6 - 2048*x^7 
+ 1024*x^8 + E^((2*x^2)/(1 + 3*x + 16*x^3))*(-16 - 80*x - 112*x^2 - 464*x^ 
3 - 1024*x^4 + 2048*x^5 - 4096*x^6 + 4096*x^7)))/(x^4 + 6*x^5 + 9*x^6 + 32 
*x^7 + 96*x^8 + 256*x^10 + E^((4*x^2)/(1 + 3*x + 16*x^3))*(16*x^2 + 96*x^3 
 + 144*x^4 + 512*x^5 + 1536*x^6 + 4096*x^8) + E^((2*x^2)/(1 + 3*x + 16*x^3 
))*(8*x^3 + 48*x^4 + 72*x^5 + 256*x^6 + 768*x^7 + 2048*x^9)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 21.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {4 \,{\mathrm e}^{x}}{x \left (4 \,{\mathrm e}^{\frac {2 x^{2}}{16 x^{3}+3 x +1}}+x \right )}\)	\(32\)
risch	\(\frac {4 \,{\mathrm e}^{x}}{x \left (4 \,{\mathrm e}^{\frac {2 x^{2}}{\left (1+4 x \right ) \left (4 x^{2}-x +1\right )}}+x \right )}\)	\(39\)

Input:

int(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*exp(2*x 
^2/(16*x^3+3*x+1))+1024*x^8-2048*x^7+384*x^6-640*x^5-220*x^4-48*x^3-44*x^2 
-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*exp(2*x^2/ 
(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp(2*x^2 
/(16*x^3+3*x+1))+256*x^10+96*x^8+32*x^7+9*x^6+6*x^5+x^4),x,method=_RETURNV 
ERBOSE)
 

Output:

4*exp(x)/x/(4*exp(2*x^2/(16*x^3+3*x+1))+x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 \, e^{x}}{x^{2} + 4 \, x e^{\left (\frac {2 \, x^{2}}{16 \, x^{3} + 3 \, x + 1}\right )}} \] Input:

integrate(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*e 
xp(2*x^2/(16*x^3+3*x+1))+1024*x^8-2048*x^7+384*x^6-640*x^5-220*x^4-48*x^3- 
44*x^2-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*exp( 
2*x^2/(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp 
(2*x^2/(16*x^3+3*x+1))+256*x^10+96*x^8+32*x^7+9*x^6+6*x^5+x^4),x, algorith 
m="fricas")
 

Output:

4*e^x/(x^2 + 4*x*e^(2*x^2/(16*x^3 + 3*x + 1)))
 

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 e^{x}}{x^{2} + 4 x e^{\frac {2 x^{2}}{16 x^{3} + 3 x + 1}}} \] Input:

integrate(((4096*x**7-4096*x**6+2048*x**5-1024*x**4-464*x**3-112*x**2-80*x 
-16)*exp(2*x**2/(16*x**3+3*x+1))+1024*x**8-2048*x**7+384*x**6-640*x**5-220 
*x**4-48*x**3-44*x**2-8*x)*exp(x)/((4096*x**8+1536*x**6+512*x**5+144*x**4+ 
96*x**3+16*x**2)*exp(2*x**2/(16*x**3+3*x+1))**2+(2048*x**9+768*x**7+256*x* 
*6+72*x**5+48*x**4+8*x**3)*exp(2*x**2/(16*x**3+3*x+1))+256*x**10+96*x**8+3 
2*x**7+9*x**6+6*x**5+x**4),x)
 

Output:

4*exp(x)/(x**2 + 4*x*exp(2*x**2/(16*x**3 + 3*x + 1)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).

Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 \, e^{\left (x + \frac {1}{12 \, {\left (4 \, x^{2} - x + 1\right )}}\right )}}{x^{2} e^{\left (\frac {1}{12 \, {\left (4 \, x^{2} - x + 1\right )}}\right )} + 4 \, x e^{\left (\frac {5 \, x}{12 \, {\left (4 \, x^{2} - x + 1\right )}} + \frac {1}{12 \, {\left (4 \, x + 1\right )}}\right )}} \] Input:

integrate(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*e 
xp(2*x^2/(16*x^3+3*x+1))+1024*x^8-2048*x^7+384*x^6-640*x^5-220*x^4-48*x^3- 
44*x^2-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*exp( 
2*x^2/(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp 
(2*x^2/(16*x^3+3*x+1))+256*x^10+96*x^8+32*x^7+9*x^6+6*x^5+x^4),x, algorith 
m="maxima")
 

Output:

4*e^(x + 1/12/(4*x^2 - x + 1))/(x^2*e^(1/12/(4*x^2 - x + 1)) + 4*x*e^(5/12 
*x/(4*x^2 - x + 1) + 1/12/(4*x + 1)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1317 vs. \(2 (29) = 58\).

Time = 0.21 (sec) , antiderivative size = 1317, normalized size of antiderivative = 41.16 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\text {Too large to display} \] Input:

integrate(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*e 
xp(2*x^2/(16*x^3+3*x+1))+1024*x^8-2048*x^7+384*x^6-640*x^5-220*x^4-48*x^3- 
44*x^2-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*exp( 
2*x^2/(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp 
(2*x^2/(16*x^3+3*x+1))+256*x^10+96*x^8+32*x^7+9*x^6+6*x^5+x^4),x, algorith 
m="giac")
 

Output:

4*(256*x^7*e^(2*x) + 1024*x^7*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) - 1024*x^ 
7*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 32*x^6*e^(2*x) - 1024* 
x^6*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 2048*x^6*e^(x + (16*x^4 + 5*x^2 + 
 x)/(16*x^3 + 3*x + 1)) + 96*x^5*e^(2*x) + 512*x^5*e^(2*x + 2*x^2/(16*x^3 
+ 3*x + 1)) - 384*x^5*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 26 
*x^4*e^(2*x) - 256*x^4*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 640*x^4*e^(x + 
 (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 5*x^3*e^(2*x) - 116*x^3*e^(2*x 
 + 2*x^2/(16*x^3 + 3*x + 1)) + 220*x^3*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^3 
 + 3*x + 1)) + 6*x^2*e^(2*x) - 28*x^2*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 
 48*x^2*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + x*e^(2*x) - 20*x 
*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 44*x*e^(x + (16*x^4 + 5*x^2 + x)/(16 
*x^3 + 3*x + 1)) - 4*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 8*e^(x + (16*x^4 
 + 5*x^2 + x)/(16*x^3 + 3*x + 1)))/(256*x^9*e^x + 1024*x^8*e^(x + 2*x^2/(1 
6*x^3 + 3*x + 1)) + 32*x^8*e^x + 1024*x^8*e^((16*x^4 + 5*x^2 + x)/(16*x^3 
+ 3*x + 1)) + 128*x^7*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + 96*x^7*e^x + 4096 
*x^7*e^(2*x^2/(16*x^3 + 3*x + 1) + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1) 
) + 128*x^7*e^((16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 384*x^6*e^(x + 2 
*x^2/(16*x^3 + 3*x + 1)) + 26*x^6*e^x + 512*x^6*e^(2*x^2/(16*x^3 + 3*x + 1 
) + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 384*x^6*e^((16*x^4 + 5*x^2 
+ x)/(16*x^3 + 3*x + 1)) + 104*x^5*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + 5...
 

Mupad [B] (verification not implemented)

Time = 2.87 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4\,{\mathrm {e}}^x}{4\,x\,{\mathrm {e}}^{\frac {2\,x^2}{16\,x^3+3\,x+1}}+x^2} \] Input:

int(-(exp(x)*(8*x + exp((2*x^2)/(3*x + 16*x^3 + 1))*(80*x + 112*x^2 + 464* 
x^3 + 1024*x^4 - 2048*x^5 + 4096*x^6 - 4096*x^7 + 16) + 44*x^2 + 48*x^3 + 
220*x^4 + 640*x^5 - 384*x^6 + 2048*x^7 - 1024*x^8))/(exp((2*x^2)/(3*x + 16 
*x^3 + 1))*(8*x^3 + 48*x^4 + 72*x^5 + 256*x^6 + 768*x^7 + 2048*x^9) + exp( 
(4*x^2)/(3*x + 16*x^3 + 1))*(16*x^2 + 96*x^3 + 144*x^4 + 512*x^5 + 1536*x^ 
6 + 4096*x^8) + x^4 + 6*x^5 + 9*x^6 + 32*x^7 + 96*x^8 + 256*x^10),x)
 

Output:

(4*exp(x))/(4*x*exp((2*x^2)/(3*x + 16*x^3 + 1)) + x^2)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 e^{x}}{x \left (4 e^{\frac {2 x^{2}}{16 x^{3}+3 x +1}}+x \right )} \] Input:

int(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*exp(2*x 
^2/(16*x^3+3*x+1))+1024*x^8-2048*x^7+384*x^6-640*x^5-220*x^4-48*x^3-44*x^2 
-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*exp(2*x^2/ 
(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp(2*x^2 
/(16*x^3+3*x+1))+256*x^10+96*x^8+32*x^7+9*x^6+6*x^5+x^4),x)
 

Output:

(4*e**x)/(x*(4*e**((2*x**2)/(16*x**3 + 3*x + 1)) + x))