3.26.12 \(\int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} (16-4 e^x)+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} (512-400 x+4 e^x x+96 x^2-8 x^3)}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx\) [2512]

3.26.12.1 Optimal result

Integrand size = 143, antiderivative size = 27 \[ \int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx=\frac {1}{4} \left (-e^x+\frac {1}{2 e^{(-4+x)^4}-x}\right )+x \]

output

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

3.26.12.2 Mathematica [F(-1)]

Timed out. \[ \int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx=\text {\$Aborted} \]

input

Integrate[(1 + E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4)*(16 - 4*E^x) + 4 
*x^2 - E^x*x^2 + E^(256 - 256*x + 96*x^2 - 16*x^3 + x^4)*(512 - 400*x + 4* 
E^x*x + 96*x^2 - 8*x^3))/(16*E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4) - 
16*E^(256 - 256*x + 96*x^2 - 16*x^3 + x^4)*x + 4*x^2),x]
 

output

$Aborted
 

3.26.12.3 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.

\(\displaystyle \int \frac {-e^x x^2+4 x^2+e^{2 x^4-32 x^3+192 x^2-512 x+512} \left (16-4 e^x\right )+e^{x^4-16 x^3+96 x^2-256 x+256} \left (-8 x^3+96 x^2+4 e^x x-400 x+512\right )+1}{4 x^2-16 e^{x^4-16 x^3+96 x^2-256 x+256} x+16 e^{2 x^4-32 x^3+192 x^2-512 x+512}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{32 x \left (x^2+16\right )} \left (-e^x x^2+4 x^2+e^{2 x^4-32 x^3+192 x^2-512 x+512} \left (16-4 e^x\right )+e^{x^4-16 x^3+96 x^2-256 x+256} \left (-8 x^3+96 x^2+4 e^x x-400 x+512\right )+1\right )}{4 \left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {e^{32 x \left (x^2+16\right )} \left (-e^x x^2+4 x^2+4 e^{2 x^4-32 x^3+192 x^2-512 x+512} \left (4-e^x\right )+4 e^{x^4-16 x^3+96 x^2-256 x+256} \left (-2 x^3+24 x^2+e^x x-100 x+128\right )+1\right )}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{4} \int \left (-\frac {8 e^{x^4-16 x^3+96 x^2+32 \left (x^2+16\right ) x-256 x+256} x^3}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}+\frac {96 e^{x^4-16 x^3+96 x^2+32 \left (x^2+16\right ) x-256 x+256} x^2}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}+\frac {4 e^{32 x \left (x^2+16\right )} x^2}{\left (e^{16 x \left (x^2+16\right )} x-2 e^{x^4+96 x^2+256}\right )^2}-\frac {e^{32 \left (x^2+16\right ) x+x} x^2}{\left (e^{16 x \left (x^2+16\right )} x-2 e^{x^4+96 x^2+256}\right )^2}-\frac {400 e^{x^4-16 x^3+96 x^2+32 \left (x^2+16\right ) x-256 x+256} x}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}+\frac {4 e^{x^4-16 x^3+96 x^2+32 \left (x^2+16\right ) x-255 x+256} x}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}+\frac {e^{32 x \left (x^2+16\right )}}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}+\frac {16 e^{2 (x-4)^4+32 x \left (x^2+16\right )}}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}+\frac {512 e^{x^4-16 x^3+96 x^2+32 \left (x^2+16\right ) x-256 x+256}}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}-\frac {4 \exp \left (2 x^4-32 x^3+192 x^2+32 \left (x^2+16\right ) x-511 x+512\right )}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (\int \frac {e^{32 x \left (x^2+16\right )}}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx+16 \int \frac {e^{2 x^4+192 x^2+512}}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx-4 \int \frac {e^{2 x^4+192 x^2+x+512}}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx+4 \int \frac {e^{32 x \left (x^2+16\right )} x^2}{\left (e^{16 x \left (x^2+16\right )} x-2 e^{x^4+96 x^2+256}\right )^2}dx+512 \int \frac {e^{x^4+16 x^3+96 x^2+256 x+256}}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx-400 \int \frac {e^{x^4+16 x^3+96 x^2+256 x+256} x}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx+4 \int \frac {e^{x^4+16 x^3+96 x^2+257 x+256} x}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx-\int \frac {e^{32 x^3+513 x} x^2}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx+96 \int \frac {e^{x^4+16 x^3+96 x^2+256 x+256} x^2}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx-8 \int \frac {e^{x^4+16 x^3+96 x^2+256 x+256} x^3}{\left (2 e^{x^4+96 x^2+256}-e^{16 x \left (x^2+16\right )} x\right )^2}dx\right )\)

input

Int[(1 + E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4)*(16 - 4*E^x) + 4*x^2 - 
 E^x*x^2 + E^(256 - 256*x + 96*x^2 - 16*x^3 + x^4)*(512 - 400*x + 4*E^x*x 
+ 96*x^2 - 8*x^3))/(16*E^(512 - 512*x + 192*x^2 - 32*x^3 + 2*x^4) - 16*E^( 
256 - 256*x + 96*x^2 - 16*x^3 + x^4)*x + 4*x^2),x]
 

output

$Aborted
 

3.26.12.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.26.12.4 Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78

method	result	size
risch	\(x -\frac {{\mathrm e}^{x}}{4}-\frac {1}{4 \left (x -2 \,{\mathrm e}^{\left (x -4\right )^{4}}\right )}\)	\(21\)
norman	\(\frac {-\frac {1}{4}+x^{2}-2 x \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}-\frac {{\mathrm e}^{x} x}{4}+\frac {{\mathrm e}^{x} {\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}{2}}{x -2 \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}\)	\(82\)
parallelrisch	\(\frac {8 x^{2}-2 \,{\mathrm e}^{x} x -16 x \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}-2+4 \,{\mathrm e}^{x} {\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}{8 x -16 \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}\)	\(85\)
parts	\(\frac {-\frac {1}{4}+x^{2}-2 x \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}{x -2 \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}+\frac {-\frac {{\mathrm e}^{x} x}{4}+\frac {{\mathrm e}^{x} {\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}{2}}{x -2 \,{\mathrm e}^{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}\)	\(110\)

input

int(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8*x^3+9 
6*x^2-400*x+512)*exp(x^4-16*x^3+96*x^2-256*x+256)-exp(x)*x^2+4*x^2+1)/(16* 
exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*x+256)+4 
*x^2),x,method=_RETURNVERBOSE)
 

output

x-1/4*exp(x)-1/4/(x-2*exp((x-4)^4))
 

3.26.12.5 Fricas [B] (verification not implemented)

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

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx=\frac {4 \, x^{2} - 2 \, {\left (4 \, x - e^{x}\right )} e^{\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )} - x e^{x} - 1}{4 \, {\left (x - 2 \, e^{\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}\right )}} \]

input

integrate(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8 
*x^3+96*x^2-400*x+512)*exp(x^4-16*x^3+96*x^2-256*x+256)-exp(x)*x^2+4*x^2+1 
)/(16*exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*x+ 
256)+4*x^2),x, algorithm=\
 

output

1/4*(4*x^2 - 2*(4*x - e^x)*e^(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) - x*e^x 
 - 1)/(x - 2*e^(x^4 - 16*x^3 + 96*x^2 - 256*x + 256))
 

3.26.12.6 Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx=x - \frac {e^{x}}{4} + \frac {1}{- 4 x + 8 e^{x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256}} \]

input

integrate(((-4*exp(x)+16)*exp(x**4-16*x**3+96*x**2-256*x+256)**2+(4*exp(x) 
*x-8*x**3+96*x**2-400*x+512)*exp(x**4-16*x**3+96*x**2-256*x+256)-exp(x)*x* 
*2+4*x**2+1)/(16*exp(x**4-16*x**3+96*x**2-256*x+256)**2-16*x*exp(x**4-16*x 
**3+96*x**2-256*x+256)+4*x**2),x)
 

output

x - exp(x)/4 + 1/(-4*x + 8*exp(x**4 - 16*x**3 + 96*x**2 - 256*x + 256))
 

3.26.12.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (20) = 40\).

Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04 \[ \int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx=-\frac {2 \, {\left (4 \, x e^{256} - e^{\left (x + 256\right )}\right )} e^{\left (x^{4} + 96 \, x^{2}\right )} + {\left (x e^{\left (257 \, x\right )} - {\left (4 \, x^{2} - 1\right )} e^{\left (256 \, x\right )}\right )} e^{\left (16 \, x^{3}\right )}}{4 \, {\left (x e^{\left (16 \, x^{3} + 256 \, x\right )} - 2 \, e^{\left (x^{4} + 96 \, x^{2} + 256\right )}\right )}} \]

input

integrate(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8 
*x^3+96*x^2-400*x+512)*exp(x^4-16*x^3+96*x^2-256*x+256)-exp(x)*x^2+4*x^2+1 
)/(16*exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*x+ 
256)+4*x^2),x, algorithm=\
 

output

-1/4*(2*(4*x*e^256 - e^(x + 256))*e^(x^4 + 96*x^2) + (x*e^(257*x) - (4*x^2 
 - 1)*e^(256*x))*e^(16*x^3))/(x*e^(16*x^3 + 256*x) - 2*e^(x^4 + 96*x^2 + 2 
56))
 

3.26.12.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 908 vs. \(2 (20) = 40\).

Time = 0.40 (sec) , antiderivative size = 908, normalized size of antiderivative = 33.63 \[ \int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx=\text {Too large to display} \]

input

integrate(((-4*exp(x)+16)*exp(x^4-16*x^3+96*x^2-256*x+256)^2+(4*exp(x)*x-8 
*x^3+96*x^2-400*x+512)*exp(x^4-16*x^3+96*x^2-256*x+256)-exp(x)*x^2+4*x^2+1 
)/(16*exp(x^4-16*x^3+96*x^2-256*x+256)^2-16*x*exp(x^4-16*x^3+96*x^2-256*x+ 
256)+4*x^2),x, algorithm=\
 

output

1/4*(16*x^7*e^x - 64*x^6*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) - 4*x^6*e 
^(2*x) - 192*x^6*e^x + 64*x^5*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) + 
 16*x^5*e^(x^4 - 16*x^3 + 96*x^2 - 254*x + 256) + 768*x^5*e^(x^4 - 16*x^3 
+ 96*x^2 - 255*x + 256) + 48*x^5*e^(2*x) + 764*x^5*e^x - 16*x^4*e^(2*x^4 - 
 32*x^3 + 192*x^2 - 510*x + 512) - 768*x^4*e^(2*x^4 - 32*x^3 + 192*x^2 - 5 
11*x + 512) - 192*x^4*e^(x^4 - 16*x^3 + 96*x^2 - 254*x + 256) - 3064*x^4*e 
^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) - 192*x^4*e^(2*x) - 976*x^4*e^x + 1 
92*x^3*e^(2*x^4 - 32*x^3 + 192*x^2 - 510*x + 512) + 3072*x^3*e^(2*x^4 - 32 
*x^3 + 192*x^2 - 511*x + 512) + 768*x^3*e^(x^4 - 16*x^3 + 96*x^2 - 254*x + 
 256) + 4000*x^3*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) + 256*x^3*e^(2*x) 
 - 196*x^3*e^x - 768*x^2*e^(2*x^4 - 32*x^3 + 192*x^2 - 510*x + 512) - 4096 
*x^2*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) - 1024*x^2*e^(x^4 - 16*x^3 
 + 96*x^2 - 254*x + 256) + 400*x^2*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) 
 + x^2*e^(2*x) + 256*x^2*e^x + 1024*x*e^(2*x^4 - 32*x^3 + 192*x^2 - 510*x 
+ 512) - 16*x*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 512) - 4*x*e^(x^4 - 16 
*x^3 + 96*x^2 - 254*x + 256) - 512*x*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 25 
6) + x*e^x + 4*e^(2*x^4 - 32*x^3 + 192*x^2 - 510*x + 512) - 2*e^(x^4 - 16* 
x^3 + 96*x^2 - 255*x + 256))/(4*x^6*e^x - 16*x^5*e^(x^4 - 16*x^3 + 96*x^2 
- 255*x + 256) - 48*x^5*e^x + 16*x^4*e^(2*x^4 - 32*x^3 + 192*x^2 - 511*x + 
 512) + 192*x^4*e^(x^4 - 16*x^3 + 96*x^2 - 255*x + 256) + 192*x^4*e^x -...
 

3.26.12.9 Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {1+e^{512-512 x+192 x^2-32 x^3+2 x^4} \left (16-4 e^x\right )+4 x^2-e^x x^2+e^{256-256 x+96 x^2-16 x^3+x^4} \left (512-400 x+4 e^x x+96 x^2-8 x^3\right )}{16 e^{512-512 x+192 x^2-32 x^3+2 x^4}-16 e^{256-256 x+96 x^2-16 x^3+x^4} x+4 x^2} \, dx=x-\frac {{\mathrm {e}}^x}{4}-\frac {1}{4\,\left (x-2\,{\mathrm {e}}^{-256\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{256}\,{\mathrm {e}}^{-16\,x^3}\,{\mathrm {e}}^{96\,x^2}\right )} \]

input

int((exp(96*x^2 - 256*x - 16*x^3 + x^4 + 256)*(4*x*exp(x) - 400*x + 96*x^2 
 - 8*x^3 + 512) - exp(192*x^2 - 512*x - 32*x^3 + 2*x^4 + 512)*(4*exp(x) - 
16) - x^2*exp(x) + 4*x^2 + 1)/(16*exp(192*x^2 - 512*x - 32*x^3 + 2*x^4 + 5 
12) - 16*x*exp(96*x^2 - 256*x - 16*x^3 + x^4 + 256) + 4*x^2),x)
 

output

x - exp(x)/4 - 1/(4*(x - 2*exp(-256*x)*exp(x^4)*exp(256)*exp(-16*x^3)*exp( 
96*x^2)))