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

Optimal. Leaf size=20 \[ -5+e^{\frac {1}{80} e^{6-\frac {6}{x}+2 x}} \]

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Rubi [F]  time = 0.59, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 18, normalized size = 0.90 \begin {gather*} e^{\frac {1}{80} e^{6-\frac {6}{x}+2 x}} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.96, size = 46, normalized size = 2.30 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.17, size = 18, normalized size = 0.90

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.54, size = 14, normalized size = 0.70 \begin {gather*} e^{\left (\frac {1}{80} \, e^{\left (2 \, x - \frac {6}{x} + 6\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.44, size = 15, normalized size = 0.75 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6\,{\mathrm {e}}^{-\frac {6}{x}}}{80}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.38, size = 15, normalized size = 0.75 \begin {gather*} e^{\frac {e^{6} e^{- \frac {2 \left (3 - x^{2}\right )}{x}}}{80}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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