3.9.5 \(\int e^{-2 x+e^{-2 x} (25 x^2+e^x (-10 x^2-10 x^3-10 e^4 x^3)+e^{2 x} (x^2+2 x^3+x^4+e^8 x^4+e^4 (2 x^3+2 x^4)))} (50 x-50 x^2+e^{2 x} (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 (6 x^2+8 x^3))+e^x (-20 x-20 x^2+10 x^3+e^4 (-30 x^2+10 x^3))) \, dx\) [805]

3.9.5.1 Optimal result

Integrand size = 167, antiderivative size = 23 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{x^2 \left (1-5 e^{-x}+x+e^4 x\right )^2} \]

output

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

3.9.5.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2} \]

input

Integrate[E^(-2*x + (25*x^2 + E^x*(-10*x^2 - 10*x^3 - 10*E^4*x^3) + E^(2*x 
)*(x^2 + 2*x^3 + x^4 + E^8*x^4 + E^4*(2*x^3 + 2*x^4)))/E^(2*x))*(50*x - 50 
*x^2 + E^(2*x)*(2*x + 6*x^2 + 4*x^3 + 4*E^8*x^3 + E^4*(6*x^2 + 8*x^3)) + E 
^x*(-20*x - 20*x^2 + 10*x^3 + E^4*(-30*x^2 + 10*x^3))),x]
 

output

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

3.9.5.3 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.

\(\displaystyle \int \left (-50 x^2+e^{2 x} \left (4 e^8 x^3+4 x^3+6 x^2+e^4 \left (8 x^3+6 x^2\right )+2 x\right )+e^x \left (10 x^3-20 x^2+e^4 \left (10 x^3-30 x^2\right )-20 x\right )+50 x\right ) \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-2 x\right ) \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-50 x^2 \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-2 x\right )+50 x \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-2 x\right )+10 \left (\left (1+e^4\right ) x^2-\left (2+3 e^4\right ) x-2\right ) x \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-x\right )+2 \left (2 \left (1+e^4\right )^2 x^2+3 \left (1+e^4\right ) x+1\right ) x \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \exp \left (e^{-2 x} x^2 \left (e^{x+4} x+e^x (x+1)-5\right )^2\right ) xdx+6 \left (1+e^4\right ) \int \exp \left (e^{-2 x} x^2 \left (e^{x+4} x+e^x (x+1)-5\right )^2\right ) x^2dx+4 \left (1+e^4\right )^2 \int \exp \left (e^{-2 x} x^2 \left (e^{x+4} x+e^x (x+1)-5\right )^2\right ) x^3dx+50 \int \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-2 x\right ) xdx-20 \int \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-x\right ) xdx-50 \int \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-2 x\right ) x^2dx-10 \left (2+3 e^4\right ) \int \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-x\right ) x^2dx+10 \left (1+e^4\right ) \int \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 e^4 x^3-10 x^3-10 x^2\right )+e^{2 x} \left (e^8 x^4+x^4+2 x^3+x^2+e^4 \left (2 x^4+2 x^3\right )\right )\right )-x\right ) x^3dx\)

input

Int[E^(-2*x + (25*x^2 + E^x*(-10*x^2 - 10*x^3 - 10*E^4*x^3) + E^(2*x)*(x^2 
 + 2*x^3 + x^4 + E^8*x^4 + E^4*(2*x^3 + 2*x^4)))/E^(2*x))*(50*x - 50*x^2 + 
 E^(2*x)*(2*x + 6*x^2 + 4*x^3 + 4*E^8*x^3 + E^4*(6*x^2 + 8*x^3)) + E^x*(-2 
0*x - 20*x^2 + 10*x^3 + E^4*(-30*x^2 + 10*x^3))),x]
 

output

$Aborted
 

3.9.5.3.1 Defintions of rubi rules used

rule 2009

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

rule 7293

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

3.9.5.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(20)=40\).

Time = 1.57 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.17

method	result	size
norman	\({\mathrm e}^{\left (\left (x^{4} {\mathrm e}^{8}+\left (2 x^{4}+2 x^{3}\right ) {\mathrm e}^{4}+x^{4}+2 x^{3}+x^{2}\right ) {\mathrm e}^{2 x}+\left (-10 x^{3} {\mathrm e}^{4}-10 x^{3}-10 x^{2}\right ) {\mathrm e}^{x}+25 x^{2}\right ) {\mathrm e}^{-2 x}}\)	\(73\)
parallelrisch	\({\mathrm e}^{\left (\left (x^{4} {\mathrm e}^{8}+\left (2 x^{4}+2 x^{3}\right ) {\mathrm e}^{4}+x^{4}+2 x^{3}+x^{2}\right ) {\mathrm e}^{2 x}+\left (-10 x^{3} {\mathrm e}^{4}-10 x^{3}-10 x^{2}\right ) {\mathrm e}^{x}+25 x^{2}\right ) {\mathrm e}^{-2 x}}\)	\(73\)
risch	\({\mathrm e}^{-x^{2} \left (-2 x^{2} {\mathrm e}^{4+2 x}-x^{2} {\mathrm e}^{2 x +8}+10 x \,{\mathrm e}^{4+x}-2 x \,{\mathrm e}^{4+2 x}-{\mathrm e}^{2 x} x^{2}+10 \,{\mathrm e}^{x} x -2 x \,{\mathrm e}^{2 x}+10 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-25\right ) {\mathrm e}^{-2 x}}\)	\(82\)

input

int(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+((10*x 
^3-30*x^2)*exp(4)+10*x^3-20*x^2-20*x)*exp(x)-50*x^2+50*x)*exp(((x^4*exp(4) 
^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10*x^3-10* 
x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x,method=_RETURNVERBOSE)
 

output

exp(((x^4*exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*e 
xp(4)-10*x^3-10*x^2)*exp(x)+25*x^2)/exp(x)^2)
 

3.9.5.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{\left ({\left (25 \, x^{2} + {\left (x^{4} e^{8} + x^{4} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{4} + x^{3}\right )} e^{4} - 2 \, x\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{3} e^{4} + x^{3} + x^{2}\right )} e^{x}\right )} e^{\left (-2 \, x\right )} + 2 \, x\right )} \]

input

integrate(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+ 
((10*x^3-30*x^2)*exp(4)+10*x^3-20*x^2-20*x)*exp(x)-50*x^2+50*x)*exp(((x^4* 
exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10*x 
^3-10*x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x, algorithm=\
 

output

e^((25*x^2 + (x^4*e^8 + x^4 + 2*x^3 + x^2 + 2*(x^4 + x^3)*e^4 - 2*x)*e^(2* 
x) - 10*(x^3*e^4 + x^3 + x^2)*e^x)*e^(-2*x) + 2*x)
 

3.9.5.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (19) = 38\).

Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{\left (25 x^{2} + \left (- 10 x^{3} e^{4} - 10 x^{3} - 10 x^{2}\right ) e^{x} + \left (x^{4} + x^{4} e^{8} + 2 x^{3} + x^{2} + \left (2 x^{4} + 2 x^{3}\right ) e^{4}\right ) e^{2 x}\right ) e^{- 2 x}} \]

input

integrate(((4*x**3*exp(4)**2+(8*x**3+6*x**2)*exp(4)+4*x**3+6*x**2+2*x)*exp 
(x)**2+((10*x**3-30*x**2)*exp(4)+10*x**3-20*x**2-20*x)*exp(x)-50*x**2+50*x 
)*exp(((x**4*exp(4)**2+(2*x**4+2*x**3)*exp(4)+x**4+2*x**3+x**2)*exp(x)**2+ 
(-10*x**3*exp(4)-10*x**3-10*x**2)*exp(x)+25*x**2)/exp(x)**2)/exp(x)**2,x)
 

output

exp((25*x**2 + (-10*x**3*exp(4) - 10*x**3 - 10*x**2)*exp(x) + (x**4 + x**4 
*exp(8) + 2*x**3 + x**2 + (2*x**4 + 2*x**3)*exp(4))*exp(2*x))*exp(-2*x))
 

3.9.5.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).

Time = 0.62 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{\left (x^{4} e^{8} + 2 \, x^{4} e^{4} + x^{4} + 2 \, x^{3} e^{4} - 10 \, x^{3} e^{\left (-x\right )} - 10 \, x^{3} e^{\left (-x + 4\right )} + 2 \, x^{3} - 10 \, x^{2} e^{\left (-x\right )} + 25 \, x^{2} e^{\left (-2 \, x\right )} + x^{2}\right )} \]

input

integrate(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+ 
((10*x^3-30*x^2)*exp(4)+10*x^3-20*x^2-20*x)*exp(x)-50*x^2+50*x)*exp(((x^4* 
exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10*x 
^3-10*x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x, algorithm=\
 

output

e^(x^4*e^8 + 2*x^4*e^4 + x^4 + 2*x^3*e^4 - 10*x^3*e^(-x) - 10*x^3*e^(-x + 
4) + 2*x^3 - 10*x^2*e^(-x) + 25*x^2*e^(-2*x) + x^2)
 

3.9.5.8 Giac [F]

\[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=\int { -2 \, {\left (25 \, x^{2} - {\left (2 \, x^{3} e^{8} + 2 \, x^{3} + 3 \, x^{2} + {\left (4 \, x^{3} + 3 \, x^{2}\right )} e^{4} + x\right )} e^{\left (2 \, x\right )} - 5 \, {\left (x^{3} - 2 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{4} - 2 \, x\right )} e^{x} - 25 \, x\right )} e^{\left ({\left (25 \, x^{2} + {\left (x^{4} e^{8} + x^{4} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{4} + x^{3}\right )} e^{4}\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{3} e^{4} + x^{3} + x^{2}\right )} e^{x}\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )} \,d x } \]

input

integrate(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+ 
((10*x^3-30*x^2)*exp(4)+10*x^3-20*x^2-20*x)*exp(x)-50*x^2+50*x)*exp(((x^4* 
exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10*x 
^3-10*x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x, algorithm=\
 

output

integrate(-2*(25*x^2 - (2*x^3*e^8 + 2*x^3 + 3*x^2 + (4*x^3 + 3*x^2)*e^4 + 
x)*e^(2*x) - 5*(x^3 - 2*x^2 + (x^3 - 3*x^2)*e^4 - 2*x)*e^x - 25*x)*e^((25* 
x^2 + (x^4*e^8 + x^4 + 2*x^3 + x^2 + 2*(x^4 + x^3)*e^4)*e^(2*x) - 10*(x^3* 
e^4 + x^3 + x^2)*e^x)*e^(-2*x) - 2*x), x)
 

3.9.5.9 Mupad [B] (verification not implemented)

Time = 8.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx={\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,x^4\,{\mathrm {e}}^4}\,{\mathrm {e}}^{x^4\,{\mathrm {e}}^8}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-10\,x^3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,x^3}\,{\mathrm {e}}^{-10\,x^2\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{-10\,x^3\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{25\,x^2\,{\mathrm {e}}^{-2\,x}} \]

input

int(exp(-2*x)*exp(exp(-2*x)*(exp(2*x)*(exp(4)*(2*x^3 + 2*x^4) + x^4*exp(8) 
 + x^2 + 2*x^3 + x^4) - exp(x)*(10*x^3*exp(4) + 10*x^2 + 10*x^3) + 25*x^2) 
)*(50*x + exp(2*x)*(2*x + exp(4)*(6*x^2 + 8*x^3) + 4*x^3*exp(8) + 6*x^2 + 
4*x^3) - exp(x)*(20*x + exp(4)*(30*x^2 - 10*x^3) + 20*x^2 - 10*x^3) - 50*x 
^2),x)
 

output

exp(2*x^3*exp(4))*exp(2*x^4*exp(4))*exp(x^4*exp(8))*exp(x^2)*exp(x^4)*exp( 
-10*x^3*exp(-x)*exp(4))*exp(2*x^3)*exp(-10*x^2*exp(-x))*exp(-10*x^3*exp(-x 
))*exp(25*x^2*exp(-2*x))