3.58.83 \(\int \frac {e^{-8+\frac {1}{9 e^8 x^2}+3 e^{\frac {1}{9 e^8 x^2}} x} (-2+9 e^8 x^2)}{3 x^2} \, dx\) [5783]

Optimal. Leaf size=17 \[ e^{3 e^{\frac {1}{9 e^8 x^2}} x} \]

[Out]

exp(3*x*exp(1/9/x^2/exp(4)^2))

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Rubi [F]
time = 0.33, 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 {e^{-8+\frac {1}{9 e^8 x^2}+3 e^{\frac {1}{9 e^8 x^2}} x} \left (-2+9 e^8 x^2\right )}{3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-8 + 1/(9*E^8*x^2) + 3*E^(1/(9*E^8*x^2))*x)*(-2 + 9*E^8*x^2))/(3*x^2),x]

[Out]

3*Defer[Int][E^(1/(9*E^8*x^2) + 3*E^(1/(9*E^8*x^2))*x), x] - (2*Defer[Int][E^(-8 + 1/(9*E^8*x^2) + 3*E^(1/(9*E
^8*x^2))*x)/x^2, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-8+\frac {1}{9 e^8 x^2}+3 e^{\frac {1}{9 e^8 x^2}} x} \left (-2+9 e^8 x^2\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (9 e^{\frac {1}{9 e^8 x^2}+3 e^{\frac {1}{9 e^8 x^2}} x}-\frac {2 e^{-8+\frac {1}{9 e^8 x^2}+3 e^{\frac {1}{9 e^8 x^2}} x}}{x^2}\right ) \, dx\\ &=-\left (\frac {2}{3} \int \frac {e^{-8+\frac {1}{9 e^8 x^2}+3 e^{\frac {1}{9 e^8 x^2}} x}}{x^2} \, dx\right )+3 \int e^{\frac {1}{9 e^8 x^2}+3 e^{\frac {1}{9 e^8 x^2}} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 17, normalized size = 1.00 \begin {gather*} e^{3 e^{\frac {1}{9 e^8 x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-8 + 1/(9*E^8*x^2) + 3*E^(1/(9*E^8*x^2))*x)*(-2 + 9*E^8*x^2))/(3*x^2),x]

[Out]

E^(3*E^(1/(9*E^8*x^2))*x)

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Maple [A]
time = 0.65, size = 13, normalized size = 0.76

method	result	size
risch	\({\mathrm e}^{3 x \,{\mathrm e}^{\frac {{\mathrm e}^{-8}}{9 x^{2}}}}\)	\(13\)
norman	\({\mathrm e}^{3 x \,{\mathrm e}^{\frac {{\mathrm e}^{-8}}{9 x^{2}}}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(9*x^2*exp(4)^2-2)*exp(1/9/x^2/exp(4)^2)*exp(3*x*exp(1/9/x^2/exp(4)^2))/x^2/exp(4)^2,x,method=_RETURNV
ERBOSE)

[Out]

exp(3*x*exp(1/9/x^2*exp(-8)))

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Maxima [A]
time = 0.55, size = 12, normalized size = 0.71 \begin {gather*} e^{\left (3 \, x e^{\left (\frac {e^{\left (-8\right )}}{9 \, x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(9*x^2*exp(4)^2-2)*exp(1/9/x^2/exp(4)^2)*exp(3*x*exp(1/9/x^2/exp(4)^2))/x^2/exp(4)^2,x, algorith
m="maxima")

[Out]

e^(3*x*e^(1/9*e^(-8)/x^2))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (12) = 24\).
time = 0.35, size = 41, normalized size = 2.41 \begin {gather*} e^{\left (\frac {{\left (27 \, x^{3} e^{\left (\frac {e^{\left (-8\right )}}{9 \, x^{2}} + 8\right )} - 72 \, x^{2} e^{8} + 1\right )} e^{\left (-8\right )}}{9 \, x^{2}} - \frac {e^{\left (-8\right )}}{9 \, x^{2}} + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(9*x^2*exp(4)^2-2)*exp(1/9/x^2/exp(4)^2)*exp(3*x*exp(1/9/x^2/exp(4)^2))/x^2/exp(4)^2,x, algorith
m="fricas")

[Out]

e^(1/9*(27*x^3*e^(1/9*e^(-8)/x^2 + 8) - 72*x^2*e^8 + 1)*e^(-8)/x^2 - 1/9*e^(-8)/x^2 + 8)

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Sympy [A]
time = 0.12, size = 15, normalized size = 0.88 \begin {gather*} e^{3 x e^{\frac {1}{9 x^{2} e^{8}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(9*x**2*exp(4)**2-2)*exp(1/9/x**2/exp(4)**2)*exp(3*x*exp(1/9/x**2/exp(4)**2))/x**2/exp(4)**2,x)

[Out]

exp(3*x*exp(exp(-8)/(9*x**2)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(9*x^2*exp(4)^2-2)*exp(1/9/x^2/exp(4)^2)*exp(3*x*exp(1/9/x^2/exp(4)^2))/x^2/exp(4)^2,x, algorith
m="giac")

[Out]

integrate(1/3*(9*x^2*e^8 - 2)*e^(3*x*e^(1/9*e^(-8)/x^2) + 1/9*e^(-8)/x^2 - 8)/x^2, x)

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Mupad [B]
time = 4.35, size = 12, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{3\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-8}}{9\,x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(-8)/(9*x^2))*exp(3*x*exp(exp(-8)/(9*x^2)))*exp(-8)*(9*x^2*exp(8) - 2))/(3*x^2),x)

[Out]

exp(3*x*exp(exp(-8)/(9*x^2)))

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