\(\int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x (-6+x+x^2)} (-2-12 x+4 x^2+6 x^3+e^x (10 x-6 x^2-2 x^3))}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x (-6+x+x^2)} x^2+8 x^3+x^4+e^x (8 x^2+2 x^3)+e^{6 x-x^2-x^3+e^x (-6+x+x^2)} (8 x^2+2 e^x x^2+2 x^3)} \, dx\) [7078]

Optimal result

Integrand size = 192, antiderivative size = 33 \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\frac {2}{x \left (4+e^x+e^{\left (-e^x+x\right ) \left (6-x-x^2\right )}+x\right )} \]

[Out]

2/(4+x+exp(x)+exp((-x^2-x+6)*(x-exp(x))))/x

Rubi [F(-1)]

Timed out. \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\text {\$Aborted} \]

[In]

Int[(-8 + E^x*(-2 - 2*x) - 4*x + E^(6*x - x^2 - x^3 + E^x*(-6 + x + x^2))*(-2 - 12*x + 4*x^2 + 6*x^3 + E^x*(10
*x - 6*x^2 - 2*x^3)))/(16*x^2 + E^(2*x)*x^2 + E^(12*x - 2*x^2 - 2*x^3 + 2*E^x*(-6 + x + x^2))*x^2 + 8*x^3 + x^
4 + E^x*(8*x^2 + 2*x^3) + E^(6*x - x^2 - x^3 + E^x*(-6 + x + x^2))*(8*x^2 + 2*E^x*x^2 + 2*x^3)),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\frac {2 e^{x^2+x^3}}{x \left (e^{x+x^2+x^3}+e^{6 x+e^x \left (-6+x+x^2\right )}+e^{x^2+x^3} (4+x)\right )} \]

[In]

Integrate[(-8 + E^x*(-2 - 2*x) - 4*x + E^(6*x - x^2 - x^3 + E^x*(-6 + x + x^2))*(-2 - 12*x + 4*x^2 + 6*x^3 + E
^x*(10*x - 6*x^2 - 2*x^3)))/(16*x^2 + E^(2*x)*x^2 + E^(12*x - 2*x^2 - 2*x^3 + 2*E^x*(-6 + x + x^2))*x^2 + 8*x^
3 + x^4 + E^x*(8*x^2 + 2*x^3) + E^(6*x - x^2 - x^3 + E^x*(-6 + x + x^2))*(8*x^2 + 2*E^x*x^2 + 2*x^3)),x]

[Out]

(2*E^(x^2 + x^3))/(x*(E^(x + x^2 + x^3) + E^(6*x + E^x*(-6 + x + x^2)) + E^(x^2 + x^3)*(4 + x)))

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82

method	result	size
risch	\(\frac {2}{x \left (x +{\mathrm e}^{x}+{\mathrm e}^{\left (3+x \right ) \left (-2+x \right ) \left ({\mathrm e}^{x}-x \right )}+4\right )}\)	\(27\)
parallelrisch	\(\frac {2}{x \left (x +{\mathrm e}^{x}+{\mathrm e}^{\left (x^{2}+x -6\right ) {\mathrm e}^{x}-x^{3}-x^{2}+6 x}+4\right )}\)	\(37\)

[In]

int((((-2*x^3-6*x^2+10*x)*exp(x)+6*x^3+4*x^2-12*x-2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+(-2-2*x)*exp(x)-4*x-8)/
(x^2*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)^2+(2*exp(x)*x^2+2*x^3+8*x^2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+exp(x)^2
*x^2+(2*x^3+8*x^2)*exp(x)+x^4+8*x^3+16*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\frac {2}{x^{2} + x e^{\left (-x^{3} - x^{2} + {\left (x^{2} + x - 6\right )} e^{x} + 6 \, x\right )} + x e^{x} + 4 \, x} \]

[In]

integrate((((-2*x^3-6*x^2+10*x)*exp(x)+6*x^3+4*x^2-12*x-2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+(-2-2*x)*exp(x)-4
*x-8)/(x^2*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)^2+(2*exp(x)*x^2+2*x^3+8*x^2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+ex
p(x)^2*x^2+(2*x^3+8*x^2)*exp(x)+x^4+8*x^3+16*x^2),x, algorithm="fricas")

[Out]

2/(x^2 + x*e^(-x^3 - x^2 + (x^2 + x - 6)*e^x + 6*x) + x*e^x + 4*x)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\frac {2}{x^{2} + x e^{x} + x e^{- x^{3} - x^{2} + 6 x + \left (x^{2} + x - 6\right ) e^{x}} + 4 x} \]

[In]

integrate((((-2*x**3-6*x**2+10*x)*exp(x)+6*x**3+4*x**2-12*x-2)*exp((x**2+x-6)*exp(x)-x**3-x**2+6*x)+(-2-2*x)*e
xp(x)-4*x-8)/(x**2*exp((x**2+x-6)*exp(x)-x**3-x**2+6*x)**2+(2*exp(x)*x**2+2*x**3+8*x**2)*exp((x**2+x-6)*exp(x)
-x**3-x**2+6*x)+exp(x)**2*x**2+(2*x**3+8*x**2)*exp(x)+x**4+8*x**3+16*x**2),x)

[Out]

2/(x**2 + x*exp(x) + x*exp(-x**3 - x**2 + 6*x + (x**2 + x - 6)*exp(x)) + 4*x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).

Time = 0.38 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\frac {2 \, e^{\left (x^{3} + x^{2} + 6 \, e^{x}\right )}}{{\left (x^{2} + x e^{x} + 4 \, x\right )} e^{\left (x^{3} + x^{2} + 6 \, e^{x}\right )} + x e^{\left (x^{2} e^{x} + x e^{x} + 6 \, x\right )}} \]

[In]

integrate((((-2*x^3-6*x^2+10*x)*exp(x)+6*x^3+4*x^2-12*x-2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+(-2-2*x)*exp(x)-4
*x-8)/(x^2*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)^2+(2*exp(x)*x^2+2*x^3+8*x^2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+ex
p(x)^2*x^2+(2*x^3+8*x^2)*exp(x)+x^4+8*x^3+16*x^2),x, algorithm="maxima")

[Out]

2*e^(x^3 + x^2 + 6*e^x)/((x^2 + x*e^x + 4*x)*e^(x^3 + x^2 + 6*e^x) + x*e^(x^2*e^x + x*e^x + 6*x))

Giac [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\frac {2}{x^{2} + x e^{\left (-x^{3} + x^{2} e^{x} - x^{2} + x e^{x} + 6 \, x - 6 \, e^{x}\right )} + x e^{x} + 4 \, x} \]

[In]

integrate((((-2*x^3-6*x^2+10*x)*exp(x)+6*x^3+4*x^2-12*x-2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+(-2-2*x)*exp(x)-4
*x-8)/(x^2*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)^2+(2*exp(x)*x^2+2*x^3+8*x^2)*exp((x^2+x-6)*exp(x)-x^3-x^2+6*x)+ex
p(x)^2*x^2+(2*x^3+8*x^2)*exp(x)+x^4+8*x^3+16*x^2),x, algorithm="giac")

[Out]

2/(x^2 + x*e^(-x^3 + x^2*e^x - x^2 + x*e^x + 6*x - 6*e^x) + x*e^x + 4*x)

Mupad [B] (verification not implemented)

Time = 16.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.94 \[ \int \frac {-8+e^x (-2-2 x)-4 x+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (-2-12 x+4 x^2+6 x^3+e^x \left (10 x-6 x^2-2 x^3\right )\right )}{16 x^2+e^{2 x} x^2+e^{12 x-2 x^2-2 x^3+2 e^x \left (-6+x+x^2\right )} x^2+8 x^3+x^4+e^x \left (8 x^2+2 x^3\right )+e^{6 x-x^2-x^3+e^x \left (-6+x+x^2\right )} \left (8 x^2+2 e^x x^2+2 x^3\right )} \, dx=\frac {{\mathrm {e}}^x\,\left (2\,x^4+8\,x^3+10\,x^2-30\,x\right )+x^2\,\left (6\,{\mathrm {e}}^{2\,x}-4\right )+x^3\,\left (2\,{\mathrm {e}}^{2\,x}-28\right )-x\,\left (10\,{\mathrm {e}}^{2\,x}-46\right )-6\,x^4}{x^2\,\left (x+{\mathrm {e}}^{6\,x-6\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x-x^2-x^3}+{\mathrm {e}}^x+4\right )\,\left (3\,x\,{\mathrm {e}}^{2\,x}-5\,{\mathrm {e}}^{2\,x}-15\,{\mathrm {e}}^x-2\,x+4\,x^2\,{\mathrm {e}}^x+x^3\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+5\,x\,{\mathrm {e}}^x-14\,x^2-3\,x^3+23\right )} \]

[In]

int(-(4*x + exp(x)*(2*x + 2) + exp(6*x - x^2 - x^3 + exp(x)*(x + x^2 - 6))*(12*x - 4*x^2 - 6*x^3 + exp(x)*(6*x
^2 - 10*x + 2*x^3) + 2) + 8)/(exp(x)*(8*x^2 + 2*x^3) + x^2*exp(12*x - 2*x^2 - 2*x^3 + 2*exp(x)*(x + x^2 - 6))
+ exp(6*x - x^2 - x^3 + exp(x)*(x + x^2 - 6))*(2*x^2*exp(x) + 8*x^2 + 2*x^3) + x^2*exp(2*x) + 16*x^2 + 8*x^3 +
x^4),x)

[Out]

(exp(x)*(10*x^2 - 30*x + 8*x^3 + 2*x^4) + x^2*(6*exp(2*x) - 4) + x^3*(2*exp(2*x) - 28) - x*(10*exp(2*x) - 46)
- 6*x^4)/(x^2*(x + exp(6*x - 6*exp(x) + x^2*exp(x) + x*exp(x) - x^2 - x^3) + exp(x) + 4)*(3*x*exp(2*x) - 5*exp
(2*x) - 15*exp(x) - 2*x + 4*x^2*exp(x) + x^3*exp(x) + x^2*exp(2*x) + 5*x*exp(x) - 14*x^2 - 3*x^3 + 23))