\(\int \frac {e^{e^{\frac {2 (3 e^x+30 x+x^2)}{x^2}}+\frac {2 (3 e^x+30 x+x^2)}{x^2}} (-60 x+e^x (-12+6 x))}{x^3} \, dx\) [6536]

Optimal result

Integrand size = 56, antiderivative size = 21 \[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx=e^{e^{2-\frac {6 \left (-10-\frac {e^x}{x}\right )}{x}}} \]

[Out]

exp(exp(1/2-3/2*(-exp(x)/x-10)/x)^4)

Rubi [F]

\[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx=\int \frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right ) \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx \]

[In]

Int[(E^(E^((2*(3*E^x + 30*x + x^2))/x^2) + (2*(3*E^x + 30*x + x^2))/x^2)*(-60*x + E^x*(-12 + 6*x)))/x^3,x]

[Out]

-12*Defer[Int][E^(E^((2*(3*E^x + 30*x + x^2))/x^2) + x + (2*(3*E^x + 30*x + x^2))/x^2)/x^3, x] - 60*Defer[Int]
[E^(E^((2*(3*E^x + 30*x + x^2))/x^2) + (2*(3*E^x + 30*x + x^2))/x^2)/x^2, x] + 6*Defer[Int][E^(E^((2*(3*E^x +
30*x + x^2))/x^2) + x + (2*(3*E^x + 30*x + x^2))/x^2)/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {6 \exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+x+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right ) (-2+x)}{x^3}-\frac {60 \exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^2}\right ) \, dx \\ & = 6 \int \frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+x+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right ) (-2+x)}{x^3} \, dx-60 \int \frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^2} \, dx \\ & = 6 \int \left (-\frac {2 \exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+x+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^3}+\frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+x+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^2}\right ) \, dx-60 \int \frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^2} \, dx \\ & = 6 \int \frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+x+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^2} \, dx-12 \int \frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+x+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^3} \, dx-60 \int \frac {\exp \left (e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}\right )}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx=e^{e^{2+\frac {6 e^x}{x^2}+\frac {60}{x}}} \]

[In]

Integrate[(E^(E^((2*(3*E^x + 30*x + x^2))/x^2) + (2*(3*E^x + 30*x + x^2))/x^2)*(-60*x + E^x*(-12 + 6*x)))/x^3,
x]

[Out]

E^E^(2 + (6*E^x)/x^2 + 60/x)

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {6 \,{\mathrm e}^{x}+2 x^{2}+60 x}{x^{2}}}}\)	\(19\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{\frac {6 \,{\mathrm e}^{x}+2 x^{2}+60 x}{x^{2}}}}\)	\(21\)

[In]

int(((6*x-12)*exp(x)-60*x)*exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4*exp(exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4)/x^3,x,me
thod=_RETURNVERBOSE)

[Out]

exp(exp(2*(3*exp(x)+x^2+30*x)/x^2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).

Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67 \[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx=e^{\left (\frac {x^{2} e^{\left (\frac {2 \, {\left (x^{2} + 30 \, x + 3 \, e^{x}\right )}}{x^{2}}\right )} + 2 \, x^{2} + 60 \, x + 6 \, e^{x}}{x^{2}} - \frac {2 \, {\left (x^{2} + 30 \, x + 3 \, e^{x}\right )}}{x^{2}}\right )} \]

[In]

integrate(((6*x-12)*exp(x)-60*x)*exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4*exp(exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4)/x^
3,x, algorithm="fricas")

[Out]

e^((x^2*e^(2*(x^2 + 30*x + 3*e^x)/x^2) + 2*x^2 + 60*x + 6*e^x)/x^2 - 2*(x^2 + 30*x + 3*e^x)/x^2)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx=e^{e^{\frac {4 \left (\frac {x^{2}}{2} + 15 x + \frac {3 e^{x}}{2}\right )}{x^{2}}}} \]

[In]

integrate(((6*x-12)*exp(x)-60*x)*exp(1/2*(3*exp(x)+x**2+30*x)/x**2)**4*exp(exp(1/2*(3*exp(x)+x**2+30*x)/x**2)*
*4)/x**3,x)

[Out]

exp(exp(4*(x**2/2 + 15*x + 3*exp(x)/2)/x**2))

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx=e^{\left (e^{\left (\frac {60}{x} + \frac {6 \, e^{x}}{x^{2}} + 2\right )}\right )} \]

[In]

integrate(((6*x-12)*exp(x)-60*x)*exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4*exp(exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4)/x^
3,x, algorithm="maxima")

[Out]

e^(e^(60/x + 6*e^x/x^2 + 2))

Giac [F]

\[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx=\int { \frac {6 \, {\left ({\left (x - 2\right )} e^{x} - 10 \, x\right )} e^{\left (\frac {2 \, {\left (x^{2} + 30 \, x + 3 \, e^{x}\right )}}{x^{2}} + e^{\left (\frac {2 \, {\left (x^{2} + 30 \, x + 3 \, e^{x}\right )}}{x^{2}}\right )}\right )}}{x^{3}} \,d x } \]

[In]

integrate(((6*x-12)*exp(x)-60*x)*exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4*exp(exp(1/2*(3*exp(x)+x^2+30*x)/x^2)^4)/x^
3,x, algorithm="giac")

[Out]

integrate(6*((x - 2)*e^x - 10*x)*e^(2*(x^2 + 30*x + 3*e^x)/x^2 + e^(2*(x^2 + 30*x + 3*e^x)/x^2))/x^3, x)

Mupad [B] (verification not implemented)

Time = 13.92 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {e^{e^{\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}}+\frac {2 \left (3 e^x+30 x+x^2\right )}{x^2}} \left (-60 x+e^x (-12+6 x)\right )}{x^3} \, dx={\mathrm {e}}^{{\mathrm {e}}^2\,{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^x}{x^2}}\,{\mathrm {e}}^{60/x}} \]

[In]

int(-(exp((4*(15*x + (3*exp(x))/2 + x^2/2))/x^2)*exp(exp((4*(15*x + (3*exp(x))/2 + x^2/2))/x^2))*(60*x - exp(x
)*(6*x - 12)))/x^3,x)

[Out]

exp(exp(2)*exp((6*exp(x))/x^2)*exp(60/x))