\(\int (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x))) \, dx\) [7507]

Optimal result

Integrand size = 76, antiderivative size = 29 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=2 e^{e^{x^2+x \left (-2+e^x-e^{6 x}+x\right )}}-x \]

[Out]

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

Rubi [F]

\[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=\int \left (-1+\exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx \]

[In]

Int[-1 + E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - 2*x + E^x*x - E^(6*x)*x + 2*x^2)*(-4 + E^(6*x)*(-2 - 12*x)
+ 8*x + E^x*(2 + 2*x)),x]

[Out]

-x - 4*Defer[Int][E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - 2*x + E^x*x - E^(6*x)*x + 2*x^2), x] + 2*Defer[Int
][E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - x + E^x*x - E^(6*x)*x + 2*x^2), x] - 2*Defer[Int][E^(E^(-2*x + E^x
*x - E^(6*x)*x + 2*x^2) + 4*x + E^x*x - E^(6*x)*x + 2*x^2), x] + 8*Defer[Int][E^(E^(-2*x + E^x*x - E^(6*x)*x +
2*x^2) - 2*x + E^x*x - E^(6*x)*x + 2*x^2)*x, x] + 2*Defer[Int][E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - x +
E^x*x - E^(6*x)*x + 2*x^2)*x, x] - 12*Defer[Int][E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) + 4*x + E^x*x - E^(6*
x)*x + 2*x^2)*x, x]

Rubi steps \begin{align*} \text {integral}& = -x+\int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right ) \, dx \\ & = -x+\int \left (-4 \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right )+8 \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) x+2 \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-x+e^x x-e^{6 x} x+2 x^2\right ) (1+x)-2 \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}+4 x+e^x x-e^{6 x} x+2 x^2\right ) (1+6 x)\right ) \, dx \\ & = -x+2 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-x+e^x x-e^{6 x} x+2 x^2\right ) (1+x) \, dx-2 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}+4 x+e^x x-e^{6 x} x+2 x^2\right ) (1+6 x) \, dx-4 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) \, dx+8 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) x \, dx \\ & = -x+2 \int \left (\exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-x+e^x x-e^{6 x} x+2 x^2\right )+\exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-x+e^x x-e^{6 x} x+2 x^2\right ) x\right ) \, dx-2 \int \left (\exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}+4 x+e^x x-e^{6 x} x+2 x^2\right )+6 \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}+4 x+e^x x-e^{6 x} x+2 x^2\right ) x\right ) \, dx-4 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) \, dx+8 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) x \, dx \\ & = -x+2 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-x+e^x x-e^{6 x} x+2 x^2\right ) \, dx-2 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}+4 x+e^x x-e^{6 x} x+2 x^2\right ) \, dx+2 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-x+e^x x-e^{6 x} x+2 x^2\right ) x \, dx-4 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) \, dx+8 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2\right ) x \, dx-12 \int \exp \left (e^{-2 x+e^x x-e^{6 x} x+2 x^2}+4 x+e^x x-e^{6 x} x+2 x^2\right ) x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=2 e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}}-x \]

[In]

Integrate[-1 + E^(E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - 2*x + E^x*x - E^(6*x)*x + 2*x^2)*(-4 + E^(6*x)*(-2 -
12*x) + 8*x + E^x*(2 + 2*x)),x]

[Out]

2*E^E^(-2*x + E^x*x - E^(6*x)*x + 2*x^2) - x

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
risch	\(2 \,{\mathrm e}^{{\mathrm e}^{x \left ({\mathrm e}^{x}-{\mathrm e}^{6 x}+2 x -2\right )}}-x\)	\(24\)
parallelrisch	\(2 \,{\mathrm e}^{{\mathrm e}^{x \left ({\mathrm e}^{x}-{\mathrm e}^{6 x}+2 x -2\right )}}-x\)	\(24\)

[In]

int(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x)*exp(exp(-x*exp(6*x)+exp(x)*x
+2*x^2-2*x))-1,x,method=_RETURNVERBOSE)

[Out]

2*exp(exp(x*(exp(x)-exp(6*x)+2*x-2)))-x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (25) = 50\).

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=-{\left (x e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )} - 2 \, e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x + e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )}\right )}\right )} e^{\left (-2 \, x^{2} + x e^{\left (6 \, x\right )} - x e^{x} + 2 \, x\right )} \]

[In]

integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x)*exp(exp(-x*exp(6*x)+ex
p(x)*x+2*x^2-2*x))-1,x, algorithm="fricas")

[Out]

-(x*e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x) - 2*e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x + e^(2*x^2 - x*e^(6*x) + x*e^x
- 2*x)))*e^(-2*x^2 + x*e^(6*x) - x*e^x + 2*x)

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=- x + 2 e^{e^{2 x^{2} - x e^{6 x} + x e^{x} - 2 x}} \]

[In]

integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x)*x+2*x**2-2*x)*exp(exp(-x*exp(6*x)+e
xp(x)*x+2*x**2-2*x))-1,x)

[Out]

-x + 2*exp(exp(2*x**2 - x*exp(6*x) + x*exp(x) - 2*x))

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=-x + 2 \, e^{\left (e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )}\right )} \]

[In]

integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x)*exp(exp(-x*exp(6*x)+ex
p(x)*x+2*x^2-2*x))-1,x, algorithm="maxima")

[Out]

-x + 2*e^(e^(2*x^2 - x*e^(6*x) + x*e^x - 2*x))

Giac [F]

\[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=\int { -2 \, {\left ({\left (6 \, x + 1\right )} e^{\left (6 \, x\right )} - {\left (x + 1\right )} e^{x} - 4 \, x + 2\right )} e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x + e^{\left (2 \, x^{2} - x e^{\left (6 \, x\right )} + x e^{x} - 2 \, x\right )}\right )} - 1 \,d x } \]

[In]

integrate(((-12*x-2)*exp(6*x)+(2+2*x)*exp(x)+8*x-4)*exp(-x*exp(6*x)+exp(x)*x+2*x^2-2*x)*exp(exp(-x*exp(6*x)+ex
p(x)*x+2*x^2-2*x))-1,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.63 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \left (-1+e^{e^{-2 x+e^x x-e^{6 x} x+2 x^2}-2 x+e^x x-e^{6 x} x+2 x^2} \left (-4+e^{6 x} (-2-12 x)+8 x+e^x (2+2 x)\right )\right ) \, dx=2\,{\mathrm {e}}^{{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{6\,x}}\,{\mathrm {e}}^{2\,x^2}}-x \]

[In]

int(exp(exp(x*exp(x) - x*exp(6*x) - 2*x + 2*x^2))*exp(x*exp(x) - x*exp(6*x) - 2*x + 2*x^2)*(8*x + exp(x)*(2*x
+ 2) - exp(6*x)*(12*x + 2) - 4) - 1,x)

[Out]

2*exp(exp(x*exp(x))*exp(-2*x)*exp(-x*exp(6*x))*exp(2*x^2)) - x