\(\int e^{-9+4 e^{e^{-9-x} x}-x} (e^{9+x} (4-4 e^x)+e^{e^{-9-x} x} (16 x-16 x^2+e^x (-16+16 x))) \, dx\) [3717]

Optimal result

Integrand size = 65, antiderivative size = 26 \[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=1+4 e^{4 e^{e^{-9-x} x}} \left (-e^x+x\right ) \]

[Out]

4*exp(exp(x/exp(x+9)))^4*(x-exp(x))+1

Rubi [F]

\[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=\int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx \]

[In]

Int[E^(-9 + 4*E^(E^(-9 - x)*x) - x)*(E^(9 + x)*(4 - 4*E^x) + E^(E^(-9 - x)*x)*(16*x - 16*x^2 + E^x*(-16 + 16*x
))),x]

[Out]

4*Defer[Int][E^(4*E^(E^(-9 - x)*x)), x] - 4*Defer[Int][E^(4*E^(E^(-9 - x)*x) + x), x] - 16*Defer[Int][E^(-9 +
4*E^(E^(-9 - x)*x) + E^(-9 - x)*x), x] + 16*Defer[Int][E^(-9 + 4*E^(E^(-9 - x)*x) + E^(-9 - x)*x)*x, x] + 16*D
efer[Int][E^(-9 + 4*E^(E^(-9 - x)*x) - x + E^(-9 - x)*x)*x, x] - 16*Defer[Int][E^(-9 + 4*E^(E^(-9 - x)*x) - x
+ E^(-9 - x)*x)*x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-4 e^{4 e^{e^{-9-x} x}} \left (-1+e^x\right )+16 e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} \left (e^x-x\right ) (-1+x)\right ) \, dx \\ & = -\left (4 \int e^{4 e^{e^{-9-x} x}} \left (-1+e^x\right ) \, dx\right )+16 \int e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} \left (e^x-x\right ) (-1+x) \, dx \\ & = -\left (4 \int \left (-e^{4 e^{e^{-9-x} x}}+e^{4 e^{e^{-9-x} x}+x}\right ) \, dx\right )+16 \int \left (e^{-9+4 e^{e^{-9-x} x}+e^{-9-x} x} (-1+x)-e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} (-1+x) x\right ) \, dx \\ & = 4 \int e^{4 e^{e^{-9-x} x}} \, dx-4 \int e^{4 e^{e^{-9-x} x}+x} \, dx+16 \int e^{-9+4 e^{e^{-9-x} x}+e^{-9-x} x} (-1+x) \, dx-16 \int e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} (-1+x) x \, dx \\ & = 4 \int e^{4 e^{e^{-9-x} x}} \, dx-4 \int e^{4 e^{e^{-9-x} x}+x} \, dx+16 \int \left (-e^{-9+4 e^{e^{-9-x} x}+e^{-9-x} x}+e^{-9+4 e^{e^{-9-x} x}+e^{-9-x} x} x\right ) \, dx-16 \int \left (-e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} x+e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} x^2\right ) \, dx \\ & = 4 \int e^{4 e^{e^{-9-x} x}} \, dx-4 \int e^{4 e^{e^{-9-x} x}+x} \, dx-16 \int e^{-9+4 e^{e^{-9-x} x}+e^{-9-x} x} \, dx+16 \int e^{-9+4 e^{e^{-9-x} x}+e^{-9-x} x} x \, dx+16 \int e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} x \, dx-16 \int e^{-9+4 e^{e^{-9-x} x}-x+e^{-9-x} x} x^2 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=e^{4 e^{e^{-9-x} x}} \left (-4 e^x+4 x\right ) \]

[In]

Integrate[E^(-9 + 4*E^(E^(-9 - x)*x) - x)*(E^(9 + x)*(4 - 4*E^x) + E^(E^(-9 - x)*x)*(16*x - 16*x^2 + E^x*(-16
+ 16*x))),x]

[Out]

E^(4*E^(E^(-9 - x)*x))*(-4*E^x + 4*x)

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
risch	\(4 \left (x -{\mathrm e}^{x}\right ) {\mathrm e}^{4 \,{\mathrm e}^{x \,{\mathrm e}^{-x -9}}}\)	\(21\)
parallelrisch	\(\left (4 \,{\mathrm e}^{4 \,{\mathrm e}^{x \,{\mathrm e}^{-x -9}}} x \,{\mathrm e}^{2 x +18}-4 \,{\mathrm e}^{4 \,{\mathrm e}^{x \,{\mathrm e}^{-x -9}}} {\mathrm e}^{x} {\mathrm e}^{2 x +18}\right ) {\mathrm e}^{-2 x -18}\)	\(52\)

[In]

int((((16*x-16)*exp(x)-16*x^2+16*x)*exp(x/exp(x+9))+(-4*exp(x)+4)*exp(x+9))*exp(exp(x/exp(x+9)))^4/exp(x+9),x,
method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=4 \, {\left (x e^{\left (x + 18\right )} - e^{\left (2 \, x + 18\right )}\right )} e^{\left (-x + 4 \, e^{\left (x e^{\left (-x - 9\right )}\right )} - 18\right )} \]

[In]

integrate((((16*x-16)*exp(x)-16*x^2+16*x)*exp(x/exp(x+9))+(-4*exp(x)+4)*exp(x+9))*exp(exp(x/exp(x+9)))^4/exp(x
+9),x, algorithm="fricas")

[Out]

4*(x*e^(x + 18) - e^(2*x + 18))*e^(-x + 4*e^(x*e^(-x - 9)) - 18)

Sympy [F(-1)]

Timed out. \[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=\text {Timed out} \]

[In]

integrate((((16*x-16)*exp(x)-16*x**2+16*x)*exp(x/exp(x+9))+(-4*exp(x)+4)*exp(x+9))*exp(exp(x/exp(x+9)))**4/exp
(x+9),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=4 \, {\left (x - e^{x}\right )} e^{\left (4 \, e^{\left (x e^{\left (-x - 9\right )}\right )}\right )} \]

[In]

integrate((((16*x-16)*exp(x)-16*x^2+16*x)*exp(x/exp(x+9))+(-4*exp(x)+4)*exp(x+9))*exp(exp(x/exp(x+9)))^4/exp(x
+9),x, algorithm="maxima")

[Out]

4*(x - e^x)*e^(4*e^(x*e^(-x - 9)))

Giac [F]

\[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=\int { -4 \, {\left (4 \, {\left (x^{2} - {\left (x - 1\right )} e^{x} - x\right )} e^{\left (x e^{\left (-x - 9\right )}\right )} + {\left (e^{x} - 1\right )} e^{\left (x + 9\right )}\right )} e^{\left (-x + 4 \, e^{\left (x e^{\left (-x - 9\right )}\right )} - 9\right )} \,d x } \]

[In]

integrate((((16*x-16)*exp(x)-16*x^2+16*x)*exp(x/exp(x+9))+(-4*exp(x)+4)*exp(x+9))*exp(exp(x/exp(x+9)))^4/exp(x
+9),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int e^{-9+4 e^{e^{-9-x} x}-x} \left (e^{9+x} \left (4-4 e^x\right )+e^{e^{-9-x} x} \left (16 x-16 x^2+e^x (-16+16 x)\right )\right ) \, dx=4\,{\mathrm {e}}^{4\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-9}}}\,\left (x-{\mathrm {e}}^x\right ) \]

[In]

int(exp(- x - 9)*exp(4*exp(x*exp(- x - 9)))*(exp(x*exp(- x - 9))*(16*x + exp(x)*(16*x - 16) - 16*x^2) - exp(x
+ 9)*(4*exp(x) - 4)),x)

[Out]

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