\(\int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} (-x^2+x^3+2 e x^3-2 e^4 x^3)}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx\) [6764]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

6*Defer[Int][E^((1 + 4*E)*x)/(3*E^(2*E*x) + E^(2*E^4*x)*x^2)^2, x] - 6*(1 - E^3)*Defer[Int][(E^(1 + (1 + 4*E)*
x)*x)/(3*E^(2*E*x) + E^(2*E^4*x)*x^2)^2, x] - Defer[Int][E^((1 + 2*E)*x)/(3*E^(2*E*x) + E^(2*E^4*x)*x^2), x] +
(1 + 2*E - 2*E^4)*Defer[Int][(E^((1 + 2*E)*x)*x)/(3*E^(2*E*x) + E^(2*E^4*x)*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 e x} \left (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 )\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx \\ & = \int \left (\frac {6 e^{x+4 e x} \left (1-e \left (1-e^3\right ) x\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2}+\frac {e^{(1-2 e) x+4 e x} \left (-1+\left (1+2 e-2 e^4\right ) x\right )}{3 e^{2 e x}+e^{2 e^4 x} x^2}\right ) \, dx \\ & = 6 \int \frac {e^{x+4 e x} \left (1-e \left (1-e^3\right ) x\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx+\int \frac {e^{(1-2 e) x+4 e x} \left (-1+\left (1+2 e-2 e^4\right ) x\right )}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx \\ & = 6 \int \frac {e^{(1+4 e) x} \left (1-e \left (1-e^3\right ) x\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx+\int \frac {e^{(1+2 e) x} \left (-1+\left (1+2 e-2 e^4\right ) x\right )}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx \\ & = 6 \int \left (\frac {e^{(1+4 e) x}}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2}+\frac {e^{1+(1+4 e) x} \left (-1+e^3\right ) x}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2}\right ) \, dx+\int \left (-\frac {e^{(1+2 e) x}}{3 e^{2 e x}+e^{2 e^4 x} x^2}-\frac {e^{(1+2 e) x} \left (-1-2 e+2 e^4\right ) x}{3 e^{2 e x}+e^{2 e^4 x} x^2}\right ) \, dx \\ & = 6 \int \frac {e^{(1+4 e) x}}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx-\left (6 \left (1-e^3\right )\right ) \int \frac {e^{1+(1+4 e) x} x}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx+\left (1+2 e-2 e^4\right ) \int \frac {e^{(1+2 e) x} x}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx-\int \frac {e^{(1+2 e) x}}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.80 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {e^{(1+2 e) x} x}{3 e^{2 e x}+e^{2 e^4 x} x^2} \]

[In]

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]

[Out]

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

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75

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

[In]

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)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {x e^{\left (2 \, x e^{4} - 2 \, x e + x\right )}}{x^{2} e^{\left (4 \, x e^{4} - 4 \, x e\right )} + 3 \, e^{\left (2 \, x e^{4} - 2 \, x e\right )}} \]

[In]

integrate(((-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, algorithm="fricas")

[Out]

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

[In]

integrate(((-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)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {x e^{\left (2 \, x e + x\right )}}{x^{2} e^{\left (2 \, x e^{4}\right )} + 3 \, e^{\left (2 \, x e\right )}} \]

[In]

integrate(((-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, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(((-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, algorithm="giac")

[Out]

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 = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, dx=\frac {x^2\,{\mathrm {e}}^x-x^3\,\left ({\mathrm {e}}^{x+1}-{\mathrm {e}}^{x+4}\right )}{\left (x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^4-2\,x\,\mathrm {e}}+3\right )\,\left (x-x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^4\right )} \]

[In]

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)

[Out]

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