\(\int \frac {e^{-e^x x+\frac {e^{-e^x x} (-361 x^2+16 e^{e^x x} x^2)}{8 x}} (-361 x+16 e^{e^x x} x+e^x (361 x^2+361 x^3))}{8 x} \, dx\) [5451]

Optimal result

Integrand size = 78, antiderivative size = 19 \[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx=e^{2 x-\frac {361}{8} e^{-e^x x} x} \]

[Out]

exp(x-361/16*x^2/exp(ln(x)+exp(x)*x))^2

Rubi [F]

\[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx=\int \frac {\exp \left (-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}\right ) \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx \]

[In]

Int[(E^(-(E^x*x) + (-361*x^2 + 16*E^(E^x*x)*x^2)/(8*E^(E^x*x)*x))*(-361*x + 16*E^(E^x*x)*x + E^x*(361*x^2 + 36
1*x^3)))/(8*x),x]

[Out]

2*Defer[Int][E^(2*x - (361*x)/(8*E^(E^x*x))), x] - (361*Defer[Int][E^(2*x - E^x*x - (361*x)/(8*E^(E^x*x))), x]
)/8 + (361*Defer[Int][E^(3*x - E^x*x - (361*x)/(8*E^(E^x*x)))*x, x])/8 + (361*Defer[Int][E^(3*x - E^x*x - (361
*x)/(8*E^(E^x*x)))*x^2, x])/8

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \int \frac {\exp \left (-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}\right ) \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{x} \, dx \\ & = \frac {1}{8} \int e^{2 x-e^x x-\frac {361}{8} e^{-e^x x} x} \left (-361+16 e^{e^x x}+361 e^x x (1+x)\right ) \, dx \\ & = \frac {1}{8} \int \left (16 e^{2 x-\frac {361}{8} e^{-e^x x} x}-361 e^{2 x-e^x x-\frac {361}{8} e^{-e^x x} x}+361 e^{3 x-e^x x-\frac {361}{8} e^{-e^x x} x} x (1+x)\right ) \, dx \\ & = 2 \int e^{2 x-\frac {361}{8} e^{-e^x x} x} \, dx-\frac {361}{8} \int e^{2 x-e^x x-\frac {361}{8} e^{-e^x x} x} \, dx+\frac {361}{8} \int e^{3 x-e^x x-\frac {361}{8} e^{-e^x x} x} x (1+x) \, dx \\ & = 2 \int e^{2 x-\frac {361}{8} e^{-e^x x} x} \, dx-\frac {361}{8} \int e^{2 x-e^x x-\frac {361}{8} e^{-e^x x} x} \, dx+\frac {361}{8} \int \left (e^{3 x-e^x x-\frac {361}{8} e^{-e^x x} x} x+e^{3 x-e^x x-\frac {361}{8} e^{-e^x x} x} x^2\right ) \, dx \\ & = 2 \int e^{2 x-\frac {361}{8} e^{-e^x x} x} \, dx-\frac {361}{8} \int e^{2 x-e^x x-\frac {361}{8} e^{-e^x x} x} \, dx+\frac {361}{8} \int e^{3 x-e^x x-\frac {361}{8} e^{-e^x x} x} x \, dx+\frac {361}{8} \int e^{3 x-e^x x-\frac {361}{8} e^{-e^x x} x} x^2 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx=e^{2 x-\frac {361}{8} e^{-e^x x} x} \]

[In]

Integrate[(E^(-(E^x*x) + (-361*x^2 + 16*E^(E^x*x)*x^2)/(8*E^(E^x*x)*x))*(-361*x + 16*E^(E^x*x)*x + E^x*(361*x^
2 + 361*x^3)))/(8*x),x]

[Out]

E^(2*x - (361*x)/(8*E^(E^x*x)))

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05

method	result	size
risch	\({\mathrm e}^{\frac {x \left (16 \,{\mathrm e}^{{\mathrm e}^{x} x}-361\right ) {\mathrm e}^{-{\mathrm e}^{x} x}}{8}}\)	\(20\)
parallelrisch	\({\mathrm e}^{\frac {\left (16 x \,{\mathrm e}^{{\mathrm e}^{x} x}-361 x \right ) {\mathrm e}^{-{\mathrm e}^{x} x}}{8}}\)	\(31\)

[In]

int(1/8*(16*exp(ln(x)+exp(x)*x)+(361*x^3+361*x^2)*exp(x)-361*x)*exp(1/16*(16*x*exp(ln(x)+exp(x)*x)-361*x^2)/ex
p(ln(x)+exp(x)*x))^2/exp(ln(x)+exp(x)*x),x,method=_RETURNVERBOSE)

[Out]

exp(1/8*x*(16*exp(exp(x)*x)-361)*exp(-exp(x)*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.47 \[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx=e^{\left (-\frac {1}{8} \, {\left (361 \, x^{2} + 8 \, {\left (x e^{x} - 2 \, x + \log \left (x\right )\right )} e^{\left (x e^{x} + \log \left (x\right )\right )}\right )} e^{\left (-x e^{x} - \log \left (x\right )\right )} + x e^{x} + \log \left (x\right )\right )} \]

[In]

integrate(1/8*(16*exp(log(x)+exp(x)*x)+(361*x^3+361*x^2)*exp(x)-361*x)*exp(1/16*(16*x*exp(log(x)+exp(x)*x)-361
*x^2)/exp(log(x)+exp(x)*x))^2/exp(log(x)+exp(x)*x),x, algorithm="fricas")

[Out]

e^(-1/8*(361*x^2 + 8*(x*e^x - 2*x + log(x))*e^(x*e^x + log(x)))*e^(-x*e^x - log(x)) + x*e^x + log(x))

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx=e^{\frac {2 \left (x^{2} e^{x e^{x}} - \frac {361 x^{2}}{16}\right ) e^{- x e^{x}}}{x}} \]

[In]

integrate(1/8*(16*exp(ln(x)+exp(x)*x)+(361*x**3+361*x**2)*exp(x)-361*x)*exp(1/16*(16*x*exp(ln(x)+exp(x)*x)-361
*x**2)/exp(ln(x)+exp(x)*x))**2/exp(ln(x)+exp(x)*x),x)

[Out]

exp(2*(x**2*exp(x*exp(x)) - 361*x**2/16)*exp(-x*exp(x))/x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx=e^{\left (-\frac {361}{8} \, x e^{\left (-x e^{x}\right )} + 2 \, x\right )} \]

[In]

integrate(1/8*(16*exp(log(x)+exp(x)*x)+(361*x^3+361*x^2)*exp(x)-361*x)*exp(1/16*(16*x*exp(log(x)+exp(x)*x)-361
*x^2)/exp(log(x)+exp(x)*x))^2/exp(log(x)+exp(x)*x),x, algorithm="maxima")

[Out]

e^(-361/8*x*e^(-x*e^x) + 2*x)

Giac [F]

\[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx=\int { \frac {1}{8} \, {\left (361 \, {\left (x^{3} + x^{2}\right )} e^{x} - 361 \, x + 16 \, e^{\left (x e^{x} + \log \left (x\right )\right )}\right )} e^{\left (-\frac {1}{8} \, {\left (361 \, x^{2} - 16 \, x e^{\left (x e^{x} + \log \left (x\right )\right )}\right )} e^{\left (-x e^{x} - \log \left (x\right )\right )} - x e^{x} - \log \left (x\right )\right )} \,d x } \]

[In]

integrate(1/8*(16*exp(log(x)+exp(x)*x)+(361*x^3+361*x^2)*exp(x)-361*x)*exp(1/16*(16*x*exp(log(x)+exp(x)*x)-361
*x^2)/exp(log(x)+exp(x)*x))^2/exp(log(x)+exp(x)*x),x, algorithm="giac")

[Out]

integrate(1/8*(361*(x^3 + x^2)*e^x - 361*x + 16*e^(x*e^x + log(x)))*e^(-1/8*(361*x^2 - 16*x*e^(x*e^x + log(x))
)*e^(-x*e^x - log(x)) - x*e^x - log(x)), x)

Mupad [B] (verification not implemented)

Time = 11.62 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-e^x x+\frac {e^{-e^x x} \left (-361 x^2+16 e^{e^x x} x^2\right )}{8 x}} \left (-361 x+16 e^{e^x x} x+e^x \left (361 x^2+361 x^3\right )\right )}{8 x} \, dx={\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-\frac {361\,x\,{\mathrm {e}}^{-x\,{\mathrm {e}}^x}}{8}} \]

[In]

int((exp(2*exp(- log(x) - x*exp(x))*(x*exp(log(x) + x*exp(x)) - (361*x^2)/16))*exp(- log(x) - x*exp(x))*(16*ex
p(log(x) + x*exp(x)) - 361*x + exp(x)*(361*x^2 + 361*x^3)))/8,x)

[Out]

exp(2*x)*exp(-(361*x*exp(-x*exp(x)))/8)