\(\int (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} (128 x^3+e^{-2+x} (-256 x^3-64 x^4)+e^{-6+3 x} (-64 x^3-48 x^4)+e^{-8+4 x} (8 x^3+8 x^4)+e^{-4+2 x} (192 x^3+96 x^4))+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} (3072 x^3+1536 x^4+(-192 x^3-96 x^4) \log (2))+e^{-8+4 x} (128 x^3+128 x^4+(-8 x^3-8 x^4) \log (2))+e^{-6+3 x} (-1024 x^3-768 x^4+(64 x^3+48 x^4) \log (2))+e^{-2+x} (-4096 x^3-1024 x^4+(256 x^3+64 x^4) \log (2)))) \, dx\) [177]

Optimal result

Integrand size = 333, antiderivative size = 27 \[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=\left (16+e^{\left (-3 x+\left (5-e^{-2+x}\right ) x\right )^4}-\log (2)\right )^2 \] Output:

(exp((x*(5-exp(-2+x))-3*x)^4)-ln(2)+16)^2
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=e^{\frac {\left (-2 e^2+e^x\right )^4 x^4}{e^8}} \left (32+e^{\frac {\left (-2 e^2+e^x\right )^4 x^4}{e^8}}-\log (4)\right ) \] Input:

Integrate[E^(32*x^4 - 64*E^(-2 + x)*x^4 + 48*E^(-4 + 2*x)*x^4 - 16*E^(-6 + 
 3*x)*x^4 + 2*E^(-8 + 4*x)*x^4)*(128*x^3 + E^(-2 + x)*(-256*x^3 - 64*x^4) 
+ E^(-6 + 3*x)*(-64*x^3 - 48*x^4) + E^(-8 + 4*x)*(8*x^3 + 8*x^4) + E^(-4 + 
 2*x)*(192*x^3 + 96*x^4)) + E^(16*x^4 - 32*E^(-2 + x)*x^4 + 24*E^(-4 + 2*x 
)*x^4 - 8*E^(-6 + 3*x)*x^4 + E^(-8 + 4*x)*x^4)*(2048*x^3 - 128*x^3*Log[2] 
+ E^(-4 + 2*x)*(3072*x^3 + 1536*x^4 + (-192*x^3 - 96*x^4)*Log[2]) + E^(-8 
+ 4*x)*(128*x^3 + 128*x^4 + (-8*x^3 - 8*x^4)*Log[2]) + E^(-6 + 3*x)*(-1024 
*x^3 - 768*x^4 + (64*x^3 + 48*x^4)*Log[2]) + E^(-2 + x)*(-4096*x^3 - 1024* 
x^4 + (256*x^3 + 64*x^4)*Log[2])),x]
 

Output:

E^(((-2*E^2 + E^x)^4*x^4)/E^8)*(32 + E^(((-2*E^2 + E^x)^4*x^4)/E^8) - Log[ 
4])
 

Rubi [A] (verified)

Time = 6.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.003, Rules used = {2009}

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.

\(\displaystyle \int \left (\left (128 x^3+e^{x-2} \left (-64 x^4-256 x^3\right )+e^{3 x-6} \left (-48 x^4-64 x^3\right )+e^{4 x-8} \left (8 x^4+8 x^3\right )+e^{2 x-4} \left (96 x^4+192 x^3\right )\right ) \exp \left (-64 e^{x-2} x^4+48 e^{2 x-4} x^4-16 e^{3 x-6} x^4+2 e^{4 x-8} x^4+32 x^4\right )+\exp \left (-32 e^{x-2} x^4+24 e^{2 x-4} x^4-8 e^{3 x-6} x^4+e^{4 x-8} x^4+16 x^4\right ) \left (2048 x^3-128 x^3 \log (2)+e^{2 x-4} \left (1536 x^4+3072 x^3+\left (-96 x^4-192 x^3\right ) \log (2)\right )+e^{4 x-8} \left (128 x^4+128 x^3+\left (-8 x^4-8 x^3\right ) \log (2)\right )+e^{3 x-6} \left (-768 x^4-1024 x^3+\left (48 x^4+64 x^3\right ) \log (2)\right )+e^{x-2} \left (-1024 x^4-4096 x^3+\left (64 x^4+256 x^3\right ) \log (2)\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle e^{\frac {2 \left (2 e^2-e^x\right )^4 x^4}{e^8}}+2 e^{\frac {\left (2 e^2-e^x\right )^4 x^4}{e^8}} (16-\log (2))\)

Input:

Int[E^(32*x^4 - 64*E^(-2 + x)*x^4 + 48*E^(-4 + 2*x)*x^4 - 16*E^(-6 + 3*x)* 
x^4 + 2*E^(-8 + 4*x)*x^4)*(128*x^3 + E^(-2 + x)*(-256*x^3 - 64*x^4) + E^(- 
6 + 3*x)*(-64*x^3 - 48*x^4) + E^(-8 + 4*x)*(8*x^3 + 8*x^4) + E^(-4 + 2*x)* 
(192*x^3 + 96*x^4)) + E^(16*x^4 - 32*E^(-2 + x)*x^4 + 24*E^(-4 + 2*x)*x^4 
- 8*E^(-6 + 3*x)*x^4 + E^(-8 + 4*x)*x^4)*(2048*x^3 - 128*x^3*Log[2] + E^(- 
4 + 2*x)*(3072*x^3 + 1536*x^4 + (-192*x^3 - 96*x^4)*Log[2]) + E^(-8 + 4*x) 
*(128*x^3 + 128*x^4 + (-8*x^3 - 8*x^4)*Log[2]) + E^(-6 + 3*x)*(-1024*x^3 - 
 768*x^4 + (64*x^3 + 48*x^4)*Log[2]) + E^(-2 + x)*(-4096*x^3 - 1024*x^4 + 
(256*x^3 + 64*x^4)*Log[2])),x]
 

Output:

E^((2*(2*E^2 - E^x)^4*x^4)/E^8) + 2*E^(((2*E^2 - E^x)^4*x^4)/E^8)*(16 - Lo 
g[2])
 

Defintions of rubi rules used

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(25)=50\).

Time = 3.60 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.26

method	result	size
parallelrisch	\(-2 \,{\mathrm e}^{x^{4} \left ({\mathrm e}^{4 x -8}-8 \,{\mathrm e}^{-6+3 x}+24 \,{\mathrm e}^{2 x -4}-32 \,{\mathrm e}^{-2+x}+16\right )} \ln \left (2\right )+{\mathrm e}^{2 x^{4} \left ({\mathrm e}^{4 x -8}-8 \,{\mathrm e}^{-6+3 x}+24 \,{\mathrm e}^{2 x -4}-32 \,{\mathrm e}^{-2+x}+16\right )}+32 \,{\mathrm e}^{x^{4} \left ({\mathrm e}^{4 x -8}-8 \,{\mathrm e}^{-6+3 x}+24 \,{\mathrm e}^{2 x -4}-32 \,{\mathrm e}^{-2+x}+16\right )}\)	\(115\)
risch	\(-2 \,{\mathrm e}^{-x^{4} \left (-{\mathrm e}^{4 x -8}+8 \,{\mathrm e}^{-6+3 x}-24 \,{\mathrm e}^{2 x -4}+32 \,{\mathrm e}^{-2+x}-16\right )} \ln \left (2\right )+{\mathrm e}^{-2 x^{4} \left (-{\mathrm e}^{4 x -8}+8 \,{\mathrm e}^{-6+3 x}-24 \,{\mathrm e}^{2 x -4}+32 \,{\mathrm e}^{-2+x}-16\right )}+32 \,{\mathrm e}^{-x^{4} \left (-{\mathrm e}^{4 x -8}+8 \,{\mathrm e}^{-6+3 x}-24 \,{\mathrm e}^{2 x -4}+32 \,{\mathrm e}^{-2+x}-16\right )}\)	\(122\)

Input:

int(((8*x^4+8*x^3)*exp(-2+x)^4+(-48*x^4-64*x^3)*exp(-2+x)^3+(96*x^4+192*x^ 
3)*exp(-2+x)^2+(-64*x^4-256*x^3)*exp(-2+x)+128*x^3)*exp(x^4*exp(-2+x)^4-8* 
x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32*x^4*exp(-2+x)+16*x^4)^2+(((-8*x^4-8* 
x^3)*ln(2)+128*x^4+128*x^3)*exp(-2+x)^4+((48*x^4+64*x^3)*ln(2)-768*x^4-102 
4*x^3)*exp(-2+x)^3+((-96*x^4-192*x^3)*ln(2)+1536*x^4+3072*x^3)*exp(-2+x)^2 
+((64*x^4+256*x^3)*ln(2)-1024*x^4-4096*x^3)*exp(-2+x)-128*x^3*ln(2)+2048*x 
^3)*exp(x^4*exp(-2+x)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32*x^4*exp(-2 
+x)+16*x^4),x,method=_RETURNVERBOSE)
 

Output:

-2*exp(x^4*(exp(-2+x)^4-8*exp(-2+x)^3+24*exp(-2+x)^2-32*exp(-2+x)+16))*ln( 
2)+exp(x^4*(exp(-2+x)^4-8*exp(-2+x)^3+24*exp(-2+x)^2-32*exp(-2+x)+16))^2+3 
2*exp(x^4*(exp(-2+x)^4-8*exp(-2+x)^3+24*exp(-2+x)^2-32*exp(-2+x)+16))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (23) = 46\).

Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.85 \[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=-2 \, {\left (\log \left (2\right ) - 16\right )} e^{\left (x^{4} e^{\left (4 \, x - 8\right )} - 8 \, x^{4} e^{\left (3 \, x - 6\right )} + 24 \, x^{4} e^{\left (2 \, x - 4\right )} - 32 \, x^{4} e^{\left (x - 2\right )} + 16 \, x^{4}\right )} + e^{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} - 16 \, x^{4} e^{\left (3 \, x - 6\right )} + 48 \, x^{4} e^{\left (2 \, x - 4\right )} - 64 \, x^{4} e^{\left (x - 2\right )} + 32 \, x^{4}\right )} \] Input:

integrate(((8*x^4+8*x^3)*exp(-2+x)^4+(-48*x^4-64*x^3)*exp(-2+x)^3+(96*x^4+ 
192*x^3)*exp(-2+x)^2+(-64*x^4-256*x^3)*exp(-2+x)+128*x^3)*exp(x^4*exp(-2+x 
)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32*x^4*exp(-2+x)+16*x^4)^2+(((-8* 
x^4-8*x^3)*log(2)+128*x^4+128*x^3)*exp(-2+x)^4+((48*x^4+64*x^3)*log(2)-768 
*x^4-1024*x^3)*exp(-2+x)^3+((-96*x^4-192*x^3)*log(2)+1536*x^4+3072*x^3)*ex 
p(-2+x)^2+((64*x^4+256*x^3)*log(2)-1024*x^4-4096*x^3)*exp(-2+x)-128*x^3*lo 
g(2)+2048*x^3)*exp(x^4*exp(-2+x)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32 
*x^4*exp(-2+x)+16*x^4),x, algorithm="fricas")
 

Output:

-2*(log(2) - 16)*e^(x^4*e^(4*x - 8) - 8*x^4*e^(3*x - 6) + 24*x^4*e^(2*x - 
4) - 32*x^4*e^(x - 2) + 16*x^4) + e^(2*x^4*e^(4*x - 8) - 16*x^4*e^(3*x - 6 
) + 48*x^4*e^(2*x - 4) - 64*x^4*e^(x - 2) + 32*x^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (20) = 40\).

Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.04 \[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=e^{- 64 x^{4} e^{x - 2} + 48 x^{4} e^{2 x - 4} - 16 x^{4} e^{3 x - 6} + 2 x^{4} e^{4 x - 8} + 32 x^{4}} + \left (32 - 2 \log {\left (2 \right )}\right ) e^{- 32 x^{4} e^{x - 2} + 24 x^{4} e^{2 x - 4} - 8 x^{4} e^{3 x - 6} + x^{4} e^{4 x - 8} + 16 x^{4}} \] Input:

integrate(((8*x**4+8*x**3)*exp(-2+x)**4+(-48*x**4-64*x**3)*exp(-2+x)**3+(9 
6*x**4+192*x**3)*exp(-2+x)**2+(-64*x**4-256*x**3)*exp(-2+x)+128*x**3)*exp( 
x**4*exp(-2+x)**4-8*x**4*exp(-2+x)**3+24*x**4*exp(-2+x)**2-32*x**4*exp(-2+ 
x)+16*x**4)**2+(((-8*x**4-8*x**3)*ln(2)+128*x**4+128*x**3)*exp(-2+x)**4+(( 
48*x**4+64*x**3)*ln(2)-768*x**4-1024*x**3)*exp(-2+x)**3+((-96*x**4-192*x** 
3)*ln(2)+1536*x**4+3072*x**3)*exp(-2+x)**2+((64*x**4+256*x**3)*ln(2)-1024* 
x**4-4096*x**3)*exp(-2+x)-128*x**3*ln(2)+2048*x**3)*exp(x**4*exp(-2+x)**4- 
8*x**4*exp(-2+x)**3+24*x**4*exp(-2+x)**2-32*x**4*exp(-2+x)+16*x**4),x)
 

Output:

exp(-64*x**4*exp(x - 2) + 48*x**4*exp(2*x - 4) - 16*x**4*exp(3*x - 6) + 2* 
x**4*exp(4*x - 8) + 32*x**4) + (32 - 2*log(2))*exp(-32*x**4*exp(x - 2) + 2 
4*x**4*exp(2*x - 4) - 8*x**4*exp(3*x - 6) + x**4*exp(4*x - 8) + 16*x**4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (23) = 46\).

Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.85 \[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=-2 \, {\left (\log \left (2\right ) - 16\right )} e^{\left (x^{4} e^{\left (4 \, x - 8\right )} - 8 \, x^{4} e^{\left (3 \, x - 6\right )} + 24 \, x^{4} e^{\left (2 \, x - 4\right )} - 32 \, x^{4} e^{\left (x - 2\right )} + 16 \, x^{4}\right )} + e^{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} - 16 \, x^{4} e^{\left (3 \, x - 6\right )} + 48 \, x^{4} e^{\left (2 \, x - 4\right )} - 64 \, x^{4} e^{\left (x - 2\right )} + 32 \, x^{4}\right )} \] Input:

integrate(((8*x^4+8*x^3)*exp(-2+x)^4+(-48*x^4-64*x^3)*exp(-2+x)^3+(96*x^4+ 
192*x^3)*exp(-2+x)^2+(-64*x^4-256*x^3)*exp(-2+x)+128*x^3)*exp(x^4*exp(-2+x 
)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32*x^4*exp(-2+x)+16*x^4)^2+(((-8* 
x^4-8*x^3)*log(2)+128*x^4+128*x^3)*exp(-2+x)^4+((48*x^4+64*x^3)*log(2)-768 
*x^4-1024*x^3)*exp(-2+x)^3+((-96*x^4-192*x^3)*log(2)+1536*x^4+3072*x^3)*ex 
p(-2+x)^2+((64*x^4+256*x^3)*log(2)-1024*x^4-4096*x^3)*exp(-2+x)-128*x^3*lo 
g(2)+2048*x^3)*exp(x^4*exp(-2+x)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32 
*x^4*exp(-2+x)+16*x^4),x, algorithm="maxima")
 

Output:

-2*(log(2) - 16)*e^(x^4*e^(4*x - 8) - 8*x^4*e^(3*x - 6) + 24*x^4*e^(2*x - 
4) - 32*x^4*e^(x - 2) + 16*x^4) + e^(2*x^4*e^(4*x - 8) - 16*x^4*e^(3*x - 6 
) + 48*x^4*e^(2*x - 4) - 64*x^4*e^(x - 2) + 32*x^4)
 

Giac [F]

\[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=\int { 8 \, {\left (16 \, x^{3} + {\left (x^{4} + x^{3}\right )} e^{\left (4 \, x - 8\right )} - 2 \, {\left (3 \, x^{4} + 4 \, x^{3}\right )} e^{\left (3 \, x - 6\right )} + 12 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{\left (2 \, x - 4\right )} - 8 \, {\left (x^{4} + 4 \, x^{3}\right )} e^{\left (x - 2\right )}\right )} e^{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} - 16 \, x^{4} e^{\left (3 \, x - 6\right )} + 48 \, x^{4} e^{\left (2 \, x - 4\right )} - 64 \, x^{4} e^{\left (x - 2\right )} + 32 \, x^{4}\right )} - 8 \, {\left (16 \, x^{3} \log \left (2\right ) - 256 \, x^{3} - {\left (16 \, x^{4} + 16 \, x^{3} - {\left (x^{4} + x^{3}\right )} \log \left (2\right )\right )} e^{\left (4 \, x - 8\right )} + 2 \, {\left (48 \, x^{4} + 64 \, x^{3} - {\left (3 \, x^{4} + 4 \, x^{3}\right )} \log \left (2\right )\right )} e^{\left (3 \, x - 6\right )} - 12 \, {\left (16 \, x^{4} + 32 \, x^{3} - {\left (x^{4} + 2 \, x^{3}\right )} \log \left (2\right )\right )} e^{\left (2 \, x - 4\right )} + 8 \, {\left (16 \, x^{4} + 64 \, x^{3} - {\left (x^{4} + 4 \, x^{3}\right )} \log \left (2\right )\right )} e^{\left (x - 2\right )}\right )} e^{\left (x^{4} e^{\left (4 \, x - 8\right )} - 8 \, x^{4} e^{\left (3 \, x - 6\right )} + 24 \, x^{4} e^{\left (2 \, x - 4\right )} - 32 \, x^{4} e^{\left (x - 2\right )} + 16 \, x^{4}\right )} \,d x } \] Input:

integrate(((8*x^4+8*x^3)*exp(-2+x)^4+(-48*x^4-64*x^3)*exp(-2+x)^3+(96*x^4+ 
192*x^3)*exp(-2+x)^2+(-64*x^4-256*x^3)*exp(-2+x)+128*x^3)*exp(x^4*exp(-2+x 
)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32*x^4*exp(-2+x)+16*x^4)^2+(((-8* 
x^4-8*x^3)*log(2)+128*x^4+128*x^3)*exp(-2+x)^4+((48*x^4+64*x^3)*log(2)-768 
*x^4-1024*x^3)*exp(-2+x)^3+((-96*x^4-192*x^3)*log(2)+1536*x^4+3072*x^3)*ex 
p(-2+x)^2+((64*x^4+256*x^3)*log(2)-1024*x^4-4096*x^3)*exp(-2+x)-128*x^3*lo 
g(2)+2048*x^3)*exp(x^4*exp(-2+x)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32 
*x^4*exp(-2+x)+16*x^4),x, algorithm="giac")
 

Output:

integrate(8*(16*x^3 + (x^4 + x^3)*e^(4*x - 8) - 2*(3*x^4 + 4*x^3)*e^(3*x - 
 6) + 12*(x^4 + 2*x^3)*e^(2*x - 4) - 8*(x^4 + 4*x^3)*e^(x - 2))*e^(2*x^4*e 
^(4*x - 8) - 16*x^4*e^(3*x - 6) + 48*x^4*e^(2*x - 4) - 64*x^4*e^(x - 2) + 
32*x^4) - 8*(16*x^3*log(2) - 256*x^3 - (16*x^4 + 16*x^3 - (x^4 + x^3)*log( 
2))*e^(4*x - 8) + 2*(48*x^4 + 64*x^3 - (3*x^4 + 4*x^3)*log(2))*e^(3*x - 6) 
 - 12*(16*x^4 + 32*x^3 - (x^4 + 2*x^3)*log(2))*e^(2*x - 4) + 8*(16*x^4 + 6 
4*x^3 - (x^4 + 4*x^3)*log(2))*e^(x - 2))*e^(x^4*e^(4*x - 8) - 8*x^4*e^(3*x 
 - 6) + 24*x^4*e^(2*x - 4) - 32*x^4*e^(x - 2) + 16*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=\int {\mathrm {e}}^{48\,x^4\,{\mathrm {e}}^{2\,x-4}-64\,x^4\,{\mathrm {e}}^{x-2}-16\,x^4\,{\mathrm {e}}^{3\,x-6}+2\,x^4\,{\mathrm {e}}^{4\,x-8}+32\,x^4}\,\left ({\mathrm {e}}^{4\,x-8}\,\left (8\,x^4+8\,x^3\right )-{\mathrm {e}}^{x-2}\,\left (64\,x^4+256\,x^3\right )-{\mathrm {e}}^{3\,x-6}\,\left (48\,x^4+64\,x^3\right )+{\mathrm {e}}^{2\,x-4}\,\left (96\,x^4+192\,x^3\right )+128\,x^3\right )-{\mathrm {e}}^{24\,x^4\,{\mathrm {e}}^{2\,x-4}-32\,x^4\,{\mathrm {e}}^{x-2}-8\,x^4\,{\mathrm {e}}^{3\,x-6}+x^4\,{\mathrm {e}}^{4\,x-8}+16\,x^4}\,\left ({\mathrm {e}}^{x-2}\,\left (4096\,x^3-\ln \left (2\right )\,\left (64\,x^4+256\,x^3\right )+1024\,x^4\right )-{\mathrm {e}}^{4\,x-8}\,\left (128\,x^3-\ln \left (2\right )\,\left (8\,x^4+8\,x^3\right )+128\,x^4\right )+{\mathrm {e}}^{3\,x-6}\,\left (1024\,x^3-\ln \left (2\right )\,\left (48\,x^4+64\,x^3\right )+768\,x^4\right )-{\mathrm {e}}^{2\,x-4}\,\left (3072\,x^3-\ln \left (2\right )\,\left (96\,x^4+192\,x^3\right )+1536\,x^4\right )+128\,x^3\,\ln \left (2\right )-2048\,x^3\right ) \,d x \] Input:

int(exp(48*x^4*exp(2*x - 4) - 64*x^4*exp(x - 2) - 16*x^4*exp(3*x - 6) + 2* 
x^4*exp(4*x - 8) + 32*x^4)*(exp(4*x - 8)*(8*x^3 + 8*x^4) - exp(x - 2)*(256 
*x^3 + 64*x^4) - exp(3*x - 6)*(64*x^3 + 48*x^4) + exp(2*x - 4)*(192*x^3 + 
96*x^4) + 128*x^3) - exp(24*x^4*exp(2*x - 4) - 32*x^4*exp(x - 2) - 8*x^4*e 
xp(3*x - 6) + x^4*exp(4*x - 8) + 16*x^4)*(exp(x - 2)*(4096*x^3 - log(2)*(2 
56*x^3 + 64*x^4) + 1024*x^4) - exp(4*x - 8)*(128*x^3 - log(2)*(8*x^3 + 8*x 
^4) + 128*x^4) + exp(3*x - 6)*(1024*x^3 - log(2)*(64*x^3 + 48*x^4) + 768*x 
^4) - exp(2*x - 4)*(3072*x^3 - log(2)*(192*x^3 + 96*x^4) + 1536*x^4) + 128 
*x^3*log(2) - 2048*x^3),x)
 

Output:

int(exp(48*x^4*exp(2*x - 4) - 64*x^4*exp(x - 2) - 16*x^4*exp(3*x - 6) + 2* 
x^4*exp(4*x - 8) + 32*x^4)*(exp(4*x - 8)*(8*x^3 + 8*x^4) - exp(x - 2)*(256 
*x^3 + 64*x^4) - exp(3*x - 6)*(64*x^3 + 48*x^4) + exp(2*x - 4)*(192*x^3 + 
96*x^4) + 128*x^3) - exp(24*x^4*exp(2*x - 4) - 32*x^4*exp(x - 2) - 8*x^4*e 
xp(3*x - 6) + x^4*exp(4*x - 8) + 16*x^4)*(exp(x - 2)*(4096*x^3 - log(2)*(2 
56*x^3 + 64*x^4) + 1024*x^4) - exp(4*x - 8)*(128*x^3 - log(2)*(8*x^3 + 8*x 
^4) + 128*x^4) + exp(3*x - 6)*(1024*x^3 - log(2)*(64*x^3 + 48*x^4) + 768*x 
^4) - exp(2*x - 4)*(3072*x^3 - log(2)*(192*x^3 + 96*x^4) + 1536*x^4) + 128 
*x^3*log(2) - 2048*x^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 6.22 \[ \int \left (e^{32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4} \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right )+e^{16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4} \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right )\right ) \, dx=\frac {e^{\frac {e^{4 x} x^{4}+24 e^{2 x} e^{4} x^{4}+16 e^{8} x^{4}}{e^{8}}} \left (e^{\frac {e^{4 x} x^{4}+24 e^{2 x} e^{4} x^{4}+16 e^{8} x^{4}}{e^{8}}}-2 e^{\frac {8 e^{3 x} x^{4}+32 e^{x} e^{4} x^{4}}{e^{6}}} \mathrm {log}\left (2\right )+32 e^{\frac {8 e^{3 x} x^{4}+32 e^{x} e^{4} x^{4}}{e^{6}}}\right )}{e^{\frac {16 e^{3 x} x^{4}+64 e^{x} e^{4} x^{4}}{e^{6}}}} \] Input:

int(((8*x^4+8*x^3)*exp(-2+x)^4+(-48*x^4-64*x^3)*exp(-2+x)^3+(96*x^4+192*x^ 
3)*exp(-2+x)^2+(-64*x^4-256*x^3)*exp(-2+x)+128*x^3)*exp(x^4*exp(-2+x)^4-8* 
x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32*x^4*exp(-2+x)+16*x^4)^2+(((-8*x^4-8* 
x^3)*log(2)+128*x^4+128*x^3)*exp(-2+x)^4+((48*x^4+64*x^3)*log(2)-768*x^4-1 
024*x^3)*exp(-2+x)^3+((-96*x^4-192*x^3)*log(2)+1536*x^4+3072*x^3)*exp(-2+x 
)^2+((64*x^4+256*x^3)*log(2)-1024*x^4-4096*x^3)*exp(-2+x)-128*x^3*log(2)+2 
048*x^3)*exp(x^4*exp(-2+x)^4-8*x^4*exp(-2+x)^3+24*x^4*exp(-2+x)^2-32*x^4*e 
xp(-2+x)+16*x^4),x)
 

Output:

(e**((e**(4*x)*x**4 + 24*e**(2*x)*e**4*x**4 + 16*e**8*x**4)/e**8)*(e**((e* 
*(4*x)*x**4 + 24*e**(2*x)*e**4*x**4 + 16*e**8*x**4)/e**8) - 2*e**((8*e**(3 
*x)*x**4 + 32*e**x*e**4*x**4)/e**6)*log(2) + 32*e**((8*e**(3*x)*x**4 + 32* 
e**x*e**4*x**4)/e**6)))/e**((16*e**(3*x)*x**4 + 64*e**x*e**4*x**4)/e**6)