\(\int e^{-e^{9 x^4}-x} (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} (-4 x+2 x^2)) \, dx\) [5276]

Optimal result

Integrand size = 56, antiderivative size = 24 \[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=e^{-x} x \left (-2 x+e^{-e^{9 x^4}} x\right ) \]

[Out]

(-2*x+x/exp(exp(9*x^4)))*x/exp(x)

Rubi [F]

\[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=\int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx \]

[In]

Int[E^(-E^(9*x^4) - x)*(2*x - x^2 - 36*E^(9*x^4)*x^5 + E^E^(9*x^4)*(-4*x + 2*x^2)),x]

[Out]

(-2*x^2)/E^x + 2*Defer[Int][E^(-E^(9*x^4) - x)*x, x] - Defer[Int][E^(-E^(9*x^4) - x)*x^2, x] - 36*Defer[Int][E
^(-E^(9*x^4) - x + 9*x^4)*x^5, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (2 e^{-e^{9 x^4}-x} x+2 e^{-x} (-2+x) x-e^{-e^{9 x^4}-x} x^2-36 e^{-e^{9 x^4}-x+9 x^4} x^5\right ) \, dx \\ & = 2 \int e^{-e^{9 x^4}-x} x \, dx+2 \int e^{-x} (-2+x) x \, dx-36 \int e^{-e^{9 x^4}-x+9 x^4} x^5 \, dx-\int e^{-e^{9 x^4}-x} x^2 \, dx \\ & = 2 \int e^{-e^{9 x^4}-x} x \, dx+2 \int \left (-2 e^{-x} x+e^{-x} x^2\right ) \, dx-36 \int e^{-e^{9 x^4}-x+9 x^4} x^5 \, dx-\int e^{-e^{9 x^4}-x} x^2 \, dx \\ & = 2 \int e^{-e^{9 x^4}-x} x \, dx+2 \int e^{-x} x^2 \, dx-4 \int e^{-x} x \, dx-36 \int e^{-e^{9 x^4}-x+9 x^4} x^5 \, dx-\int e^{-e^{9 x^4}-x} x^2 \, dx \\ & = 4 e^{-x} x-2 e^{-x} x^2+2 \int e^{-e^{9 x^4}-x} x \, dx-4 \int e^{-x} \, dx+4 \int e^{-x} x \, dx-36 \int e^{-e^{9 x^4}-x+9 x^4} x^5 \, dx-\int e^{-e^{9 x^4}-x} x^2 \, dx \\ & = 4 e^{-x}-2 e^{-x} x^2+2 \int e^{-e^{9 x^4}-x} x \, dx+4 \int e^{-x} \, dx-36 \int e^{-e^{9 x^4}-x+9 x^4} x^5 \, dx-\int e^{-e^{9 x^4}-x} x^2 \, dx \\ & = -2 e^{-x} x^2+2 \int e^{-e^{9 x^4}-x} x \, dx-36 \int e^{-e^{9 x^4}-x+9 x^4} x^5 \, dx-\int e^{-e^{9 x^4}-x} x^2 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=-e^{-e^{9 x^4}-x} \left (-1+2 e^{e^{9 x^4}}\right ) x^2 \]

[In]

Integrate[E^(-E^(9*x^4) - x)*(2*x - x^2 - 36*E^(9*x^4)*x^5 + E^E^(9*x^4)*(-4*x + 2*x^2)),x]

[Out]

-(E^(-E^(9*x^4) - x)*(-1 + 2*E^E^(9*x^4))*x^2)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17

method	result	size
risch	\(-2 x^{2} {\mathrm e}^{-x}+x^{2} {\mathrm e}^{-x -{\mathrm e}^{9 x^{4}}}\)	\(28\)
parallelrisch	\(\left (-2 \,{\mathrm e}^{{\mathrm e}^{9 x^{4}}} x^{2}+x^{2}\right ) {\mathrm e}^{-x} {\mathrm e}^{-{\mathrm e}^{9 x^{4}}}\)	\(31\)

[In]

int(((2*x^2-4*x)*exp(exp(9*x^4))-36*x^5*exp(9*x^4)-x^2+2*x)/exp(x)/exp(exp(9*x^4)),x,method=_RETURNVERBOSE)

[Out]

-2*x^2*exp(-x)+x^2*exp(-x-exp(9*x^4))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=-2 \, x^{2} e^{\left (-x\right )} + x^{2} e^{\left (-x - e^{\left (9 \, x^{4}\right )}\right )} \]

[In]

integrate(((2*x^2-4*x)*exp(exp(9*x^4))-36*x^5*exp(9*x^4)-x^2+2*x)/exp(x)/exp(exp(9*x^4)),x, algorithm="fricas"
)

[Out]

-2*x^2*e^(-x) + x^2*e^(-x - e^(9*x^4))

Sympy [A] (verification not implemented)

Time = 2.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=- 2 x^{2} e^{- x} + x^{2} e^{- x} e^{- e^{9 x^{4}}} \]

[In]

integrate(((2*x**2-4*x)*exp(exp(9*x**4))-36*x**5*exp(9*x**4)-x**2+2*x)/exp(x)/exp(exp(9*x**4)),x)

[Out]

-2*x**2*exp(-x) + x**2*exp(-x)*exp(-exp(9*x**4))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=x^{2} e^{\left (-x - e^{\left (9 \, x^{4}\right )}\right )} - 2 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + 4 \, {\left (x + 1\right )} e^{\left (-x\right )} \]

[In]

integrate(((2*x^2-4*x)*exp(exp(9*x^4))-36*x^5*exp(9*x^4)-x^2+2*x)/exp(x)/exp(exp(9*x^4)),x, algorithm="maxima"
)

[Out]

x^2*e^(-x - e^(9*x^4)) - 2*(x^2 + 2*x + 2)*e^(-x) + 4*(x + 1)*e^(-x)

Giac [F]

\[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=\int { -{\left (36 \, x^{5} e^{\left (9 \, x^{4}\right )} + x^{2} - 2 \, {\left (x^{2} - 2 \, x\right )} e^{\left (e^{\left (9 \, x^{4}\right )}\right )} - 2 \, x\right )} e^{\left (-x - e^{\left (9 \, x^{4}\right )}\right )} \,d x } \]

[In]

integrate(((2*x^2-4*x)*exp(exp(9*x^4))-36*x^5*exp(9*x^4)-x^2+2*x)/exp(x)/exp(exp(9*x^4)),x, algorithm="giac")

[Out]

integrate(-(36*x^5*e^(9*x^4) + x^2 - 2*(x^2 - 2*x)*e^(e^(9*x^4)) - 2*x)*e^(-x - e^(9*x^4)), x)

Mupad [B] (verification not implemented)

Time = 14.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int e^{-e^{9 x^4}-x} \left (2 x-x^2-36 e^{9 x^4} x^5+e^{e^{9 x^4}} \left (-4 x+2 x^2\right )\right ) \, dx=x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-{\mathrm {e}}^{9\,x^4}}-2\,x^2\,{\mathrm {e}}^{-x} \]

[In]

int(-exp(-x)*exp(-exp(9*x^4))*(exp(exp(9*x^4))*(4*x - 2*x^2) - 2*x + 36*x^5*exp(9*x^4) + x^2),x)

[Out]

x^2*exp(-x)*exp(-exp(9*x^4)) - 2*x^2*exp(-x)