3.9.96 \(\int -\frac {e^{4+\frac {1}{3} (-9-e^4 \log (x)+3 \log ^2(\log (4)))}}{3 x} \, dx\)

Optimal. Leaf size=18 \[ e^{-3-\frac {1}{3} e^4 \log (x)+\log ^2(\log (4))} \]

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Rubi [A]  time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 2203} \begin {gather*} x^{-\frac {e^4}{3}} e^{\log ^2(\log (4))-3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1/3*E^(4 + (-9 - E^4*Log[x] + 3*Log[Log[4]]^2)/3)/x,x]

[Out]

E^(-3 + Log[Log[4]]^2)/x^(E^4/3)

Rule 12

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

Rule 2203

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[F^(a + b*(c + d*x))/(b*d*Log[F]), x] /; FreeQ
[{F, a, b, c, d}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\frac {1}{3} \int \frac {\exp \left (4+\frac {1}{3} \left (-9-e^4 \log (x)+3 \log ^2(\log (4))\right )\right )}{x} \, dx\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int e^{4+\frac {1}{3} \left (-9-e^4 x+3 \log ^2(\log (4))\right )} \, dx,x,\log (x)\right )\right )\\ &=e^{-3+\log ^2(\log (4))} x^{-\frac {e^4}{3}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.06 \begin {gather*} e^{-3+\log ^2(\log (4))} x^{-\frac {e^4}{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1/3*E^(4 + (-9 - E^4*Log[x] + 3*Log[Log[4]]^2)/3)/x,x]

[Out]

E^(-3 + Log[Log[4]]^2)/x^(E^4/3)

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fricas [A]  time = 0.74, size = 16, normalized size = 0.89 \begin {gather*} e^{\left (-\frac {1}{3} \, e^{4} \log \relax (x) + \log \left (2 \, \log \relax (2)\right )^{2} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3*exp(2)^2*exp(log(2*log(2))^2-1/3*exp(2)^2*log(x)-3)/x,x, algorithm="fricas")

[Out]

e^(-1/3*e^4*log(x) + log(2*log(2))^2 - 3)

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giac [A]  time = 0.36, size = 16, normalized size = 0.89 \begin {gather*} e^{\left (-\frac {1}{3} \, e^{4} \log \relax (x) + \log \left (2 \, \log \relax (2)\right )^{2} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1/3*e^4*log(x) + log(2*log(2))^2 - 3)

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maple [A]  time = 0.11, size = 19, normalized size = 1.06

method	result	size
gosper	\({\mathrm e}^{\ln \left (2 \ln \relax (2)\right )^{2}-\frac {{\mathrm e}^{4} \ln \relax (x )}{3}-3}\)	\(19\)
derivativedivides	\({\mathrm e}^{\ln \left (2 \ln \relax (2)\right )^{2}-\frac {{\mathrm e}^{4} \ln \relax (x )}{3}-3}\)	\(19\)
default	\({\mathrm e}^{\ln \left (2 \ln \relax (2)\right )^{2}-\frac {{\mathrm e}^{4} \ln \relax (x )}{3}-3}\)	\(19\)
norman	\({\mathrm e}^{\ln \left (2 \ln \relax (2)\right )^{2}-\frac {{\mathrm e}^{4} \ln \relax (x )}{3}-3}\)	\(19\)
risch	\(x^{-\frac {{\mathrm e}^{4}}{3}} \ln \relax (2)^{2 \ln \relax (2)} {\mathrm e}^{\ln \relax (2)^{2}-3+\ln \left (\ln \relax (2)\right )^{2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 16, normalized size = 0.89 \begin {gather*} e^{\left (-\frac {1}{3} \, e^{4} \log \relax (x) + \log \left (2 \, \log \relax (2)\right )^{2} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/3*exp(2)^2*exp(log(2*log(2))^2-1/3*exp(2)^2*log(x)-3)/x,x, algorithm="maxima")

[Out]

e^(-1/3*e^4*log(x) + log(2*log(2))^2 - 3)

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mupad [B]  time = 0.67, size = 29, normalized size = 1.61 \begin {gather*} \frac {2^{2\,\ln \left (\ln \relax (2)\right )}\,{\mathrm {e}}^{{\ln \relax (2)}^2}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{{\ln \left (\ln \relax (2)\right )}^2}}{x^{\frac {{\mathrm {e}}^4}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(2*log(log(2)))*exp(log(2)^2)*exp(-3)*exp(log(log(2))^2))/x^(exp(4)/3)

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sympy [A]  time = 0.30, size = 19, normalized size = 1.06 \begin {gather*} \frac {1}{x^{\frac {e^{4}}{3}} e^{3 - \log {\left (2 \log {\relax (2 )} \right )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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