\(\int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 (3-x^2)}{x}}-\frac {2 (3-x^2)}{x}} (3+x^2)}{40 x^2} \, dx\) [123]

Optimal result

Integrand size = 48, antiderivative size = 20 \[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx=-5+e^{\frac {1}{80} e^{6-\frac {6}{x}+2 x}} \]

[Out]

exp(1/80*exp(3)^2/exp(3/x-x)^2)-5

Rubi [F]

\[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx=\int \frac {\exp \left (6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}\right ) \left (3+x^2\right )}{40 x^2} \, dx \]

[In]

Int[(E^(6 + E^(6 - (2*(3 - x^2))/x)/80 - (2*(3 - x^2))/x)*(3 + x^2))/(40*x^2),x]

[Out]

Defer[Int][E^(6 + E^(6 - (2*(3 - x^2))/x)/80 - (2*(3 - x^2))/x), x]/40 + (3*Defer[Int][E^(6 + E^(6 - (2*(3 - x
^2))/x)/80 - (2*(3 - x^2))/x)/x^2, x])/40

Rubi steps \begin{align*} \text {integral}& = \frac {1}{40} \int \frac {\exp \left (6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}\right ) \left (3+x^2\right )}{x^2} \, dx \\ & = \frac {1}{40} \int \left (\exp \left (6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}\right )+\frac {3 \exp \left (6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}\right )}{x^2}\right ) \, dx \\ & = \frac {1}{40} \int \exp \left (6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}\right ) \, dx+\frac {3}{40} \int \frac {\exp \left (6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}\right )}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx=e^{\frac {1}{80} e^{6-\frac {6}{x}+2 x}} \]

[In]

Integrate[(E^(6 + E^(6 - (2*(3 - x^2))/x)/80 - (2*(3 - x^2))/x)*(3 + x^2))/(40*x^2),x]

[Out]

E^(E^(6 - 6/x + 2*x)/80)

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

method	result	size
risch	\({\mathrm e}^{\frac {{\mathrm e}^{\frac {2 x^{2}+6 x -6}{x}}}{80}}\)	\(18\)
parallelrisch	\({\mathrm e}^{\frac {{\mathrm e}^{6} {\mathrm e}^{\frac {2 x^{2}-6}{x}}}{80}}\)	\(21\)
derivativedivides	\({\mathrm e}^{\frac {{\mathrm e}^{6} {\mathrm e}^{-\frac {2 \left (-x^{2}+3\right )}{x}}}{80}}\)	\(22\)
default	\({\mathrm e}^{\frac {{\mathrm e}^{6} {\mathrm e}^{-\frac {2 \left (-x^{2}+3\right )}{x}}}{80}}\)	\(22\)
norman	\({\mathrm e}^{\frac {{\mathrm e}^{6} {\mathrm e}^{-\frac {2 \left (-x^{2}+3\right )}{x}}}{80}}\)	\(22\)

[In]

int(1/40*(x^2+3)*exp(3)^2*exp(1/80*exp(3)^2/exp((-x^2+3)/x)^2)/x^2/exp((-x^2+3)/x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(1/80*exp(2*(x^2+3*x-3)/x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (16) = 32\).

Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx=e^{\left (\frac {160 \, x^{2} + x e^{\left (\frac {2 \, {\left (x^{2} + 3 \, x - 3\right )}}{x}\right )} + 480 \, x - 480}{80 \, x} - \frac {2 \, {\left (x^{2} + 3 \, x - 3\right )}}{x}\right )} \]

[In]

integrate(1/40*(x^2+3)*exp(3)^2*exp(1/80*exp(3)^2/exp((-x^2+3)/x)^2)/x^2/exp((-x^2+3)/x)^2,x, algorithm="frica
s")

[Out]

e^(1/80*(160*x^2 + x*e^(2*(x^2 + 3*x - 3)/x) + 480*x - 480)/x - 2*(x^2 + 3*x - 3)/x)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx=e^{\frac {e^{6} e^{- \frac {2 \cdot \left (3 - x^{2}\right )}{x}}}{80}} \]

[In]

integrate(1/40*(x**2+3)*exp(3)**2*exp(1/80*exp(3)**2/exp((-x**2+3)/x)**2)/x**2/exp((-x**2+3)/x)**2,x)

[Out]

exp(exp(6)*exp(-2*(3 - x**2)/x)/80)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx=e^{\left (\frac {1}{80} \, e^{\left (2 \, x - \frac {6}{x} + 6\right )}\right )} \]

[In]

integrate(1/40*(x^2+3)*exp(3)^2*exp(1/80*exp(3)^2/exp((-x^2+3)/x)^2)/x^2/exp((-x^2+3)/x)^2,x, algorithm="maxim
a")

[Out]

e^(1/80*e^(2*x - 6/x + 6))

Giac [F]

\[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx=\int { \frac {{\left (x^{2} + 3\right )} e^{\left (\frac {2 \, {\left (x^{2} - 3\right )}}{x} + \frac {1}{80} \, e^{\left (\frac {2 \, {\left (x^{2} - 3\right )}}{x} + 6\right )} + 6\right )}}{40 \, x^{2}} \,d x } \]

[In]

integrate(1/40*(x^2+3)*exp(3)^2*exp(1/80*exp(3)^2/exp((-x^2+3)/x)^2)/x^2/exp((-x^2+3)/x)^2,x, algorithm="giac"
)

[Out]

integrate(1/40*(x^2 + 3)*e^(2*(x^2 - 3)/x + 1/80*e^(2*(x^2 - 3)/x + 6) + 6)/x^2, x)

Mupad [B] (verification not implemented)

Time = 8.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {e^{6+\frac {1}{80} e^{6-\frac {2 \left (3-x^2\right )}{x}}-\frac {2 \left (3-x^2\right )}{x}} \left (3+x^2\right )}{40 x^2} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6\,{\mathrm {e}}^{-\frac {6}{x}}}{80}} \]

[In]

int((exp(6)*exp((2*(x^2 - 3))/x)*exp((exp(6)*exp((2*(x^2 - 3))/x))/80)*(x^2 + 3))/(40*x^2),x)

[Out]

exp((exp(2*x)*exp(6)*exp(-6/x))/80)