\(\int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} (-x^2+x^3+2 e x^3-2 e^4 x^3)}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx\) [764]

Optimal result

Integrand size = 88, antiderivative size = 32 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=1+\frac {e^x x}{3+e^{-2 x+2 \left (1-e+e^4\right ) x} x^2} \] Output:

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

Mathematica [A] (verified)

Time = 3.99 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {e^{(1+2 e) x} x}{3 e^{2 e x}+e^{2 e^4 x} x^2} \] Input:

Integrate[(E^x*(3 + 3*x) + E^(x - 2*E*x + 2*E^4*x)*(-x^2 + x^3 + 2*E*x^3 - 
 2*E^4*x^3))/(9 + 6*E^(-2*E*x + 2*E^4*x)*x^2 + E^(-4*E*x + 4*E^4*x)*x^4),x 
]
 

Output:

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

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.

\(\displaystyle \int \frac {e^{2 e^4 x-2 e x+x} \left (-2 e^4 x^3+2 e x^3+x^3-x^2\right )+e^x (3 x+3)}{e^{4 e^4 x-4 e x} x^4+6 e^{2 e^4 x-2 e x} x^2+9} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{4 e x} \left (e^{2 e^4 x-2 e x+x} \left (-2 e^4 x^3+2 e x^3+x^3-x^2\right )+e^x (3 x+3)\right )}{\left (e^{2 e^4 x} x^2+3 e^{2 e x}\right )^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(E^x*(3 + 3*x) + E^(x - 2*E*x + 2*E^4*x)*(-x^2 + x^3 + 2*E*x^3 - 2*E^4 
*x^3))/(9 + 6*E^(-2*E*x + 2*E^4*x)*x^2 + E^(-4*E*x + 4*E^4*x)*x^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75

method	result	size
risch	\(\frac {x \,{\mathrm e}^{x}}{x^{2} {\mathrm e}^{-2 x \left (-{\mathrm e}^{4}+{\mathrm e}\right )}+3}\)	\(24\)
parallelrisch	\(\frac {{\mathrm e}^{x} x}{x^{2} {\mathrm e}^{2 x \left ({\mathrm e}^{4}-{\mathrm e}\right )}+3}\)	\(25\)

Input:

int(((-2*x^3*exp(4)+2*x^3*exp(1)+x^3-x^2)*exp(x)*exp(x*exp(4)-x*exp(1))^2+ 
(3*x+3)*exp(x))/(x^4*exp(x*exp(4)-x*exp(1))^4+6*x^2*exp(x*exp(4)-x*exp(1)) 
^2+9),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {x e^{\left (2 \, x e^{4} - 2 \, x e + x\right )}}{x^{2} e^{\left (4 \, x e^{4} - 4 \, x e\right )} + 3 \, e^{\left (2 \, x e^{4} - 2 \, x e\right )}} \] Input:

integrate(((-2*x^3*exp(4)+2*x^3*exp(1)+x^3-x^2)*exp(x)*exp(x*exp(4)-exp(1) 
*x)^2+(3*x+3)*exp(x))/(x^4*exp(x*exp(4)-exp(1)*x)^4+6*x^2*exp(x*exp(4)-exp 
(1)*x)^2+9),x, algorithm="fricas")
 

Output:

x*e^(2*x*e^4 - 2*x*e + x)/(x^2*e^(4*x*e^4 - 4*x*e) + 3*e^(2*x*e^4 - 2*x*e) 
)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\text {Timed out} \] Input:

integrate(((-2*x**3*exp(4)+2*x**3*exp(1)+x**3-x**2)*exp(x)*exp(x*exp(4)-ex 
p(1)*x)**2+(3*x+3)*exp(x))/(x**4*exp(x*exp(4)-exp(1)*x)**4+6*x**2*exp(x*ex 
p(4)-exp(1)*x)**2+9),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {x e^{\left (2 \, x e + x\right )}}{x^{2} e^{\left (2 \, x e^{4}\right )} + 3 \, e^{\left (2 \, x e\right )}} \] Input:

integrate(((-2*x^3*exp(4)+2*x^3*exp(1)+x^3-x^2)*exp(x)*exp(x*exp(4)-exp(1) 
*x)^2+(3*x+3)*exp(x))/(x^4*exp(x*exp(4)-exp(1)*x)^4+6*x^2*exp(x*exp(4)-exp 
(1)*x)^2+9),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\int { -\frac {{\left (2 \, x^{3} e^{4} - 2 \, x^{3} e - x^{3} + x^{2}\right )} e^{\left (2 \, x e^{4} - 2 \, x e + x\right )} - 3 \, {\left (x + 1\right )} e^{x}}{x^{4} e^{\left (4 \, x e^{4} - 4 \, x e\right )} + 6 \, x^{2} e^{\left (2 \, x e^{4} - 2 \, x e\right )} + 9} \,d x } \] Input:

integrate(((-2*x^3*exp(4)+2*x^3*exp(1)+x^3-x^2)*exp(x)*exp(x*exp(4)-exp(1) 
*x)^2+(3*x+3)*exp(x))/(x^4*exp(x*exp(4)-exp(1)*x)^4+6*x^2*exp(x*exp(4)-exp 
(1)*x)^2+9),x, algorithm="giac")
 

Output:

integrate(-((2*x^3*e^4 - 2*x^3*e - x^3 + x^2)*e^(2*x*e^4 - 2*x*e + x) - 3* 
(x + 1)*e^x)/(x^4*e^(4*x*e^4 - 4*x*e) + 6*x^2*e^(2*x*e^4 - 2*x*e) + 9), x)
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {x^2\,{\mathrm {e}}^x-x^3\,\left ({\mathrm {e}}^{x+1}-{\mathrm {e}}^{x+4}\right )}{\left (x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^4-2\,x\,\mathrm {e}}+3\right )\,\left (x-x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^4\right )} \] Input:

int((exp(x)*(3*x + 3) + exp(2*x*exp(4) - 2*x*exp(1))*exp(x)*(2*x^3*exp(1) 
- 2*x^3*exp(4) - x^2 + x^3))/(6*x^2*exp(2*x*exp(4) - 2*x*exp(1)) + x^4*exp 
(4*x*exp(4) - 4*x*exp(1)) + 9),x)
 

Output:

(x^2*exp(x) - x^3*(exp(x + 1) - exp(x + 4)))/((x^2*exp(2*x*exp(4) - 2*x*ex 
p(1)) + 3)*(x - x^2*exp(1) + x^2*exp(4)))
 

Reduce [F]

\[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=3 \left (\int \frac {e^{4 e x +x}}{e^{4 e^{4} x} x^{4}+6 e^{2 e^{4} x +2 e x} x^{2}+9 e^{4 e x}}d x \right )+3 \left (\int \frac {e^{4 e x +x} x}{e^{4 e^{4} x} x^{4}+6 e^{2 e^{4} x +2 e x} x^{2}+9 e^{4 e x}}d x \right )-2 \left (\int \frac {e^{2 e^{4} x +2 e x +x} x^{3}}{e^{4 e^{4} x} x^{4}+6 e^{2 e^{4} x +2 e x} x^{2}+9 e^{4 e x}}d x \right ) e^{4}+2 \left (\int \frac {e^{2 e^{4} x +2 e x +x} x^{3}}{e^{4 e^{4} x} x^{4}+6 e^{2 e^{4} x +2 e x} x^{2}+9 e^{4 e x}}d x \right ) e +\int \frac {e^{2 e^{4} x +2 e x +x} x^{3}}{e^{4 e^{4} x} x^{4}+6 e^{2 e^{4} x +2 e x} x^{2}+9 e^{4 e x}}d x -\left (\int \frac {e^{2 e^{4} x +2 e x +x} x^{2}}{e^{4 e^{4} x} x^{4}+6 e^{2 e^{4} x +2 e x} x^{2}+9 e^{4 e x}}d x \right ) \] Input:

int(((-2*x^3*exp(4)+2*x^3*exp(1)+x^3-x^2)*exp(x)*exp(x*exp(4)-exp(1)*x)^2+ 
(3*x+3)*exp(x))/(x^4*exp(x*exp(4)-exp(1)*x)^4+6*x^2*exp(x*exp(4)-exp(1)*x) 
^2+9),x)
 

Output:

3*int(e**(4*e*x + x)/(e**(4*e**4*x)*x**4 + 6*e**(2*e**4*x + 2*e*x)*x**2 + 
9*e**(4*e*x)),x) + 3*int((e**(4*e*x + x)*x)/(e**(4*e**4*x)*x**4 + 6*e**(2* 
e**4*x + 2*e*x)*x**2 + 9*e**(4*e*x)),x) - 2*int((e**(2*e**4*x + 2*e*x + x) 
*x**3)/(e**(4*e**4*x)*x**4 + 6*e**(2*e**4*x + 2*e*x)*x**2 + 9*e**(4*e*x)), 
x)*e**4 + 2*int((e**(2*e**4*x + 2*e*x + x)*x**3)/(e**(4*e**4*x)*x**4 + 6*e 
**(2*e**4*x + 2*e*x)*x**2 + 9*e**(4*e*x)),x)*e + int((e**(2*e**4*x + 2*e*x 
 + x)*x**3)/(e**(4*e**4*x)*x**4 + 6*e**(2*e**4*x + 2*e*x)*x**2 + 9*e**(4*e 
*x)),x) - int((e**(2*e**4*x + 2*e*x + x)*x**2)/(e**(4*e**4*x)*x**4 + 6*e** 
(2*e**4*x + 2*e*x)*x**2 + 9*e**(4*e*x)),x)