3.60.73 \(\int \frac {e^{-\frac {2}{3} (1-2 x+2 x^2)} (-12+(4 x-8 x^2) \log (x^2))}{3 x \log ^3(x^2)} \, dx\) [5973]

Optimal. Leaf size=27 \[ \frac {e^{-\frac {2}{3} x \left (\frac {(1-x)^2}{x}+x\right )}}{\log ^2\left (x^2\right )} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 40, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 2326} \begin {gather*} \frac {e^{-\frac {2}{3} \left (2 x^2-2 x+1\right )} \left (x-2 x^2\right )}{(1-2 x) x \log ^2\left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 + (4*x - 8*x^2)*Log[x^2])/(3*E^((2*(1 - 2*x + 2*x^2))/3)*x*Log[x^2]^3),x]

[Out]

(x - 2*x^2)/(E^((2*(1 - 2*x + 2*x^2))/3)*(1 - 2*x)*x*Log[x^2]^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 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-\frac {2}{3} \left (1-2 x+2 x^2\right )} \left (-12+\left (4 x-8 x^2\right ) \log \left (x^2\right )\right )}{x \log ^3\left (x^2\right )} \, dx\\ &=\frac {e^{-\frac {2}{3} \left (1-2 x+2 x^2\right )} \left (x-2 x^2\right )}{(1-2 x) x \log ^2\left (x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.85 \begin {gather*} \frac {e^{-\frac {2}{3} \left (1-2 x+2 x^2\right )}}{\log ^2\left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 + (4*x - 8*x^2)*Log[x^2])/(3*E^((2*(1 - 2*x + 2*x^2))/3)*x*Log[x^2]^3),x]

[Out]

1/(E^((2*(1 - 2*x + 2*x^2))/3)*Log[x^2]^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.28, size = 66, normalized size = 2.44

method	result	size
risch	\(-\frac {4 \,{\mathrm e}^{-\frac {4}{3} x^{2}+\frac {4}{3} x -\frac {2}{3}}}{\left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )\right )^{2}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-8*x^2+4*x)*ln(x^2)-12)/x/exp(2/3*x^2-2/3*x+1/3)^2/ln(x^2)^3,x,method=_RETURNVERBOSE)

[Out]

-4/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+4*I*ln(x))^2*exp(-4/3*x^2+4/3*x-2
/3)

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Maxima [A]
time = 0.33, size = 17, normalized size = 0.63 \begin {gather*} \frac {e^{\left (-\frac {4}{3} \, x^{2} + \frac {4}{3} \, x - \frac {2}{3}\right )}}{4 \, \log \left (x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-8*x^2+4*x)*log(x^2)-12)/x/exp(2/3*x^2-2/3*x+1/3)^2/log(x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 18, normalized size = 0.67 \begin {gather*} \frac {e^{\left (-\frac {4}{3} \, x^{2} + \frac {4}{3} \, x - \frac {2}{3}\right )}}{\log \left (x^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-8*x^2+4*x)*log(x^2)-12)/x/exp(2/3*x^2-2/3*x+1/3)^2/log(x^2)^3,x, algorithm="fricas")

[Out]

e^(-4/3*x^2 + 4/3*x - 2/3)/log(x^2)^2

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Sympy [A]
time = 0.08, size = 22, normalized size = 0.81 \begin {gather*} \frac {e^{- \frac {4 x^{2}}{3} + \frac {4 x}{3} - \frac {2}{3}}}{\log {\left (x^{2} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-8*x**2+4*x)*ln(x**2)-12)/x/exp(2/3*x**2-2/3*x+1/3)**2/ln(x**2)**3,x)

[Out]

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

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Giac [A]
time = 0.40, size = 18, normalized size = 0.67 \begin {gather*} \frac {e^{\left (-\frac {4}{3} \, x^{2} + \frac {4}{3} \, x - \frac {2}{3}\right )}}{\log \left (x^{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-8*x^2+4*x)*log(x^2)-12)/x/exp(2/3*x^2-2/3*x+1/3)^2/log(x^2)^3,x, algorithm="giac")

[Out]

e^(-4/3*x^2 + 4/3*x - 2/3)/log(x^2)^2

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Mupad [B]
time = 4.39, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^{\frac {4\,x}{3}}\,{\mathrm {e}}^{-\frac {2}{3}}\,{\mathrm {e}}^{-\frac {4\,x^2}{3}}}{{\ln \left (x^2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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