3.38.49 \(\int \frac {e^{\frac {6 x^2-x^4-\log (3)}{x^2}} (2 x^4-2 \log (3))}{x^3 \log (3)} \, dx\)

Optimal. Leaf size=29 \[ \frac {-3-e^{6-x^2-\frac {\log (3)}{x^2}}+5 \log (3)}{\log (3)} \]

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Rubi [A]  time = 0.24, antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 6706} \begin {gather*} -\frac {3^{-\frac {1}{x^2}} e^{\frac {6 x^2-x^4}{x^2}}}{\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((6*x^2 - x^4 - Log[3])/x^2)*(2*x^4 - 2*Log[3]))/(x^3*Log[3]),x]

[Out]

-(E^((6*x^2 - x^4)/x^2)/(3^x^(-2)*Log[3]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{\frac {6 x^2-x^4-\log (3)}{x^2}} \left (2 x^4-2 \log (3)\right )}{x^3} \, dx}{\log (3)}\\ &=-\frac {3^{-\frac {1}{x^2}} e^{\frac {6 x^2-x^4}{x^2}}}{\log (3)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 22, normalized size = 0.76 \begin {gather*} -\frac {3^{-\frac {1}{x^2}} e^{6-x^2}}{\log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((6*x^2 - x^4 - Log[3])/x^2)*(2*x^4 - 2*Log[3]))/(x^3*Log[3]),x]

[Out]

-(E^(6 - x^2)/(3^x^(-2)*Log[3]))

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fricas [A]  time = 0.68, size = 23, normalized size = 0.79 \begin {gather*} -\frac {e^{\left (-\frac {x^{4} - 6 \, x^{2} + \log \relax (3)}{x^{2}}\right )}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(3)+2*x^4)*exp((-log(3)-x^4+6*x^2)/x^2)/x^3/log(3),x, algorithm="fricas")

[Out]

-e^(-(x^4 - 6*x^2 + log(3))/x^2)/log(3)

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giac [A]  time = 0.14, size = 21, normalized size = 0.72 \begin {gather*} -\frac {e^{\left (-x^{2} - \frac {\log \relax (3)}{x^{2}} + 6\right )}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(3)+2*x^4)*exp((-log(3)-x^4+6*x^2)/x^2)/x^3/log(3),x, algorithm="giac")

[Out]

-e^(-x^2 - log(3)/x^2 + 6)/log(3)

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maple [A]  time = 0.08, size = 20, normalized size = 0.69

method	result	size
risch	\(-\frac {\left (\frac {1}{3}\right )^{\frac {1}{x^{2}}} {\mathrm e}^{-x^{2}+6}}{\ln \relax (3)}\)	\(20\)
gosper	\(-\frac {{\mathrm e}^{-\frac {x^{4}-6 x^{2}+\ln \relax (3)}{x^{2}}}}{\ln \relax (3)}\)	\(24\)
norman	\(-\frac {{\mathrm e}^{\frac {-\ln \relax (3)-x^{4}+6 x^{2}}{x^{2}}}}{\ln \relax (3)}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(3)+2*x^4)*exp((-ln(3)-x^4+6*x^2)/x^2)/x^3/ln(3),x,method=_RETURNVERBOSE)

[Out]

-(1/3)^(1/x^2)*exp(-x^2+6)/ln(3)

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maxima [A]  time = 0.52, size = 21, normalized size = 0.72 \begin {gather*} -\frac {e^{\left (-x^{2} - \frac {\log \relax (3)}{x^{2}} + 6\right )}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(3)+2*x^4)*exp((-log(3)-x^4+6*x^2)/x^2)/x^3/log(3),x, algorithm="maxima")

[Out]

-e^(-x^2 - log(3)/x^2 + 6)/log(3)

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mupad [B]  time = 2.43, size = 21, normalized size = 0.72 \begin {gather*} -\frac {{\mathrm {e}}^6\,{\mathrm {e}}^{-x^2}}{3^{\frac {1}{x^2}}\,\ln \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(log(3) - 6*x^2 + x^4)/x^2)*(2*log(3) - 2*x^4))/(x^3*log(3)),x)

[Out]

-(exp(6)*exp(-x^2))/(3^(1/x^2)*log(3))

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sympy [A]  time = 0.16, size = 20, normalized size = 0.69 \begin {gather*} - \frac {e^{\frac {- x^{4} + 6 x^{2} - \log {\relax (3 )}}{x^{2}}}}{\log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(3)+2*x**4)*exp((-ln(3)-x**4+6*x**2)/x**2)/x**3/ln(3),x)

[Out]

-exp((-x**4 + 6*x**2 - log(3))/x**2)/log(3)

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