3.93.45 \(\int \frac {e^{-\frac {(-4-x^2) \log ^2(3)+400 \log ^4(x)}{\log ^2(3)}} (8 x^2 \log ^2(3)-6400 \log ^3(x))}{x \log ^2(3)} \, dx\)

Optimal. Leaf size=19 \[ 4 e^{4+x^2-\frac {400 \log ^4(x)}{\log ^2(3)}} \]

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Rubi [A]  time = 0.29, antiderivative size = 26, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {12, 6706} \begin {gather*} 4 e^{\frac {\left (x^2+4\right ) \log ^2(3)-400 \log ^4(x)}{\log ^2(3)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*x^2*Log[3]^2 - 6400*Log[x]^3)/(E^(((-4 - x^2)*Log[3]^2 + 400*Log[x]^4)/Log[3]^2)*x*Log[3]^2),x]

[Out]

4*E^(((4 + x^2)*Log[3]^2 - 400*Log[x]^4)/Log[3]^2)

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 {\exp \left (-\frac {\left (-4-x^2\right ) \log ^2(3)+400 \log ^4(x)}{\log ^2(3)}\right ) \left (8 x^2 \log ^2(3)-6400 \log ^3(x)\right )}{x} \, dx}{\log ^2(3)}\\ &=4 e^{\frac {\left (4+x^2\right ) \log ^2(3)-400 \log ^4(x)}{\log ^2(3)}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 19, normalized size = 1.00 \begin {gather*} 4 e^{4+x^2-\frac {400 \log ^4(x)}{\log ^2(3)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*x^2*Log[3]^2 - 6400*Log[x]^3)/(E^(((-4 - x^2)*Log[3]^2 + 400*Log[x]^4)/Log[3]^2)*x*Log[3]^2),x]

[Out]

4*E^(4 + x^2 - (400*Log[x]^4)/Log[3]^2)

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fricas [A]  time = 0.62, size = 27, normalized size = 1.42 \begin {gather*} 4 \, e^{\left (-\frac {400 \, \log \relax (x)^{4} - {\left (x^{2} + 4\right )} \log \relax (3)^{2}}{\log \relax (3)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6400*log(x)^3+8*x^2*log(3)^2)/x/log(3)^2/exp((400*log(x)^4+(-x^2-4)*log(3)^2)/log(3)^2),x, algorit
hm="fricas")

[Out]

4*e^(-(400*log(x)^4 - (x^2 + 4)*log(3)^2)/log(3)^2)

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giac [A]  time = 1.78, size = 18, normalized size = 0.95 \begin {gather*} 4 \, e^{\left (x^{2} - \frac {400 \, \log \relax (x)^{4}}{\log \relax (3)^{2}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6400*log(x)^3+8*x^2*log(3)^2)/x/log(3)^2/exp((400*log(x)^4+(-x^2-4)*log(3)^2)/log(3)^2),x, algorit
hm="giac")

[Out]

4*e^(x^2 - 400*log(x)^4/log(3)^2 + 4)

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maple [A]  time = 0.04, size = 30, normalized size = 1.58

method	result	size
risch	\(4 \,{\mathrm e}^{\frac {-400 \ln \relax (x )^{4}+x^{2} \ln \relax (3)^{2}+4 \ln \relax (3)^{2}}{\ln \relax (3)^{2}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6400*ln(x)^3+8*x^2*ln(3)^2)/x/ln(3)^2/exp((400*ln(x)^4+(-x^2-4)*ln(3)^2)/ln(3)^2),x,method=_RETURNVERBOS
E)

[Out]

4*exp((-400*ln(x)^4+x^2*ln(3)^2+4*ln(3)^2)/ln(3)^2)

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maxima [A]  time = 0.59, size = 18, normalized size = 0.95 \begin {gather*} 4 \, e^{\left (x^{2} - \frac {400 \, \log \relax (x)^{4}}{\log \relax (3)^{2}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6400*log(x)^3+8*x^2*log(3)^2)/x/log(3)^2/exp((400*log(x)^4+(-x^2-4)*log(3)^2)/log(3)^2),x, algorit
hm="maxima")

[Out]

4*e^(x^2 - 400*log(x)^4/log(3)^2 + 4)

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mupad [B]  time = 7.30, size = 19, normalized size = 1.00 \begin {gather*} 4\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-\frac {400\,{\ln \relax (x)}^4}{{\ln \relax (3)}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(400*log(x)^4 - log(3)^2*(x^2 + 4))/log(3)^2)*(8*x^2*log(3)^2 - 6400*log(x)^3))/(x*log(3)^2),x)

[Out]

4*exp(x^2)*exp(4)*exp(-(400*log(x)^4)/log(3)^2)

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sympy [A]  time = 0.36, size = 26, normalized size = 1.37 \begin {gather*} 4 e^{- \frac {\left (- x^{2} - 4\right ) \log {\relax (3 )}^{2} + 400 \log {\relax (x )}^{4}}{\log {\relax (3 )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6400*ln(x)**3+8*x**2*ln(3)**2)/x/ln(3)**2/exp((400*ln(x)**4+(-x**2-4)*ln(3)**2)/ln(3)**2),x)

[Out]

4*exp(-((-x**2 - 4)*log(3)**2 + 400*log(x)**4)/log(3)**2)

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