\(\int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} (e^x (2+2 x)+e^{x+e^{4 x} x} (e^x (-1-2 x)+e^{5 x} (-x-4 x^2))) \, dx\) [1882]

Optimal result

Integrand size = 73, antiderivative size = 31 \[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=4-e^{1-e^x \left (2 x-e^{x+e^{4 x} x} x\right )} \]

[Out]

4-exp(1-exp(x)*(2*x-x*exp(x*exp(4*x)+x)))

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6838} \[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=-e^{-2 e^x x+e^{e^{4 x} x+2 x} x+1} \]

[In]

Int[E^(1 - 2*E^x*x + E^(2*x + E^(4*x)*x)*x)*(E^x*(2 + 2*x) + E^(x + E^(4*x)*x)*(E^x*(-1 - 2*x) + E^(5*x)*(-x -
4*x^2))),x]

[Out]

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

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = -e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=-e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \]

[In]

Integrate[E^(1 - 2*E^x*x + E^(2*x + E^(4*x)*x)*x)*(E^x*(2 + 2*x) + E^(x + E^(4*x)*x)*(E^x*(-1 - 2*x) + E^(5*x)
*(-x - 4*x^2))),x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71

method	result	size
risch	\(-{\mathrm e}^{x \,{\mathrm e}^{x \left (2+{\mathrm e}^{4 x}\right )}-2 \,{\mathrm e}^{x} x +1}\)	\(22\)
parallelrisch	\(-{\mathrm e}^{x \,{\mathrm e}^{x} {\mathrm e}^{x \left ({\mathrm e}^{4 x}+1\right )}-2 \,{\mathrm e}^{x} x +1}\)	\(24\)

[In]

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

[Out]

-exp(x*exp(x*(2+exp(4*x)))-2*exp(x)*x+1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=-e^{\left (x e^{\left (x e^{\left (4 \, x\right )} + 2 \, x\right )} - 2 \, x e^{x} + 1\right )} \]

[In]

integrate((((-4*x^2-x)*exp(x)*exp(4*x)+(-1-2*x)*exp(x))*exp(x*exp(4*x)+x)+(2+2*x)*exp(x))*exp(x*exp(x)*exp(x*e
xp(4*x)+x)-2*exp(x)*x+1),x, algorithm="fricas")

[Out]

-e^(x*e^(x*e^(4*x) + 2*x) - 2*x*e^x + 1)

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=- e^{x e^{x} e^{x e^{4 x} + x} - 2 x e^{x} + 1} \]

[In]

integrate((((-4*x**2-x)*exp(x)*exp(4*x)+(-1-2*x)*exp(x))*exp(x*exp(4*x)+x)+(2+2*x)*exp(x))*exp(x*exp(x)*exp(x*
exp(4*x)+x)-2*exp(x)*x+1),x)

[Out]

-exp(x*exp(x)*exp(x*exp(4*x) + x) - 2*x*exp(x) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=-e^{\left (x e^{\left (x e^{\left (4 \, x\right )} + 2 \, x\right )} - 2 \, x e^{x} + 1\right )} \]

[In]

integrate((((-4*x^2-x)*exp(x)*exp(4*x)+(-1-2*x)*exp(x))*exp(x*exp(4*x)+x)+(2+2*x)*exp(x))*exp(x*exp(x)*exp(x*e
xp(4*x)+x)-2*exp(x)*x+1),x, algorithm="maxima")

[Out]

-e^(x*e^(x*e^(4*x) + 2*x) - 2*x*e^x + 1)

Giac [F]

\[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=\int { -{\left ({\left ({\left (4 \, x^{2} + x\right )} e^{\left (5 \, x\right )} + {\left (2 \, x + 1\right )} e^{x}\right )} e^{\left (x e^{\left (4 \, x\right )} + x\right )} - 2 \, {\left (x + 1\right )} e^{x}\right )} e^{\left (x e^{\left (x e^{\left (4 \, x\right )} + 2 \, x\right )} - 2 \, x e^{x} + 1\right )} \,d x } \]

[In]

integrate((((-4*x^2-x)*exp(x)*exp(4*x)+(-1-2*x)*exp(x))*exp(x*exp(4*x)+x)+(2+2*x)*exp(x))*exp(x*exp(x)*exp(x*e
xp(4*x)+x)-2*exp(x)*x+1),x, algorithm="giac")

[Out]

integrate(-(((4*x^2 + x)*e^(5*x) + (2*x + 1)*e^x)*e^(x*e^(4*x) + x) - 2*(x + 1)*e^x)*e^(x*e^(x*e^(4*x) + 2*x)
- 2*x*e^x + 1), x)

Mupad [B] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int e^{1-2 e^x x+e^{2 x+e^{4 x} x} x} \left (e^x (2+2 x)+e^{x+e^{4 x} x} \left (e^x (-1-2 x)+e^{5 x} \left (-x-4 x^2\right )\right )\right ) \, dx=-{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^x}\,\mathrm {e}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{4\,x}}} \]

[In]

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

[Out]

-exp(-2*x*exp(x))*exp(1)*exp(x*exp(2*x)*exp(x*exp(4*x)))