3.5.59 \(\int \frac {4-4 x^2-8 x^2 \log (x)}{(x \log (\frac {3}{5 x})-x^3 \log (x)+x \log (x^2)) \log (-\log (\frac {3}{5 x})+x^2 \log (x)-\log (x^2))} \, dx\)

Optimal. Leaf size=29 \[ 4 \log \left (3 \log \left (-\log \left (\frac {3}{5 x}\right )+x^2 \log (x)-\log \left (x^2\right )\right )\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 27, normalized size of antiderivative = 0.93, number of steps used = 1, number of rules used = 1, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6684} \begin {gather*} 4 \log \left (\log \left (x^2 \log (x)-\log \left (x^2\right )-\log \left (\frac {3}{5 x}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 4*x^2 - 8*x^2*Log[x])/((x*Log[3/(5*x)] - x^3*Log[x] + x*Log[x^2])*Log[-Log[3/(5*x)] + x^2*Log[x] - Lo
g[x^2]]),x]

[Out]

4*Log[Log[-Log[3/(5*x)] + x^2*Log[x] - Log[x^2]]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 \log \left (\log \left (-\log \left (\frac {3}{5 x}\right )+x^2 \log (x)-\log \left (x^2\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 27, normalized size = 0.93 \begin {gather*} 4 \log \left (\log \left (-\log \left (\frac {3}{5 x}\right )+x^2 \log (x)-\log \left (x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 4*x^2 - 8*x^2*Log[x])/((x*Log[3/(5*x)] - x^3*Log[x] + x*Log[x^2])*Log[-Log[3/(5*x)] + x^2*Log[x
] - Log[x^2]]),x]

[Out]

4*Log[Log[-Log[3/(5*x)] + x^2*Log[x] - Log[x^2]]]

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fricas [A]  time = 0.65, size = 27, normalized size = 0.93 \begin {gather*} 4 \, \log \left (\log \left (\frac {1}{2} \, {\left (x^{2} - 2\right )} \log \left (\frac {9}{25}\right ) - {\left (x^{2} - 1\right )} \log \left (\frac {3}{5 \, x}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^2*log(x)-4*x^2+4)/(x*log(x^2)-x^3*log(x)+x*log(3/5/x))/log(-log(x^2)+x^2*log(x)-log(3/5/x)),x,
algorithm="fricas")

[Out]

4*log(log(1/2*(x^2 - 2)*log(9/25) - (x^2 - 1)*log(3/5/x)))

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giac [A]  time = 0.76, size = 21, normalized size = 0.72 \begin {gather*} 4 \, \log \left (\log \left (x^{2} \log \relax (x) + \log \relax (5) - \log \relax (3) - \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^2*log(x)-4*x^2+4)/(x*log(x^2)-x^3*log(x)+x*log(3/5/x))/log(-log(x^2)+x^2*log(x)-log(3/5/x)),x,
algorithm="giac")

[Out]

4*log(log(x^2*log(x) + log(5) - log(3) - log(x)))

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maple [C]  time = 0.25, size = 50, normalized size = 1.72

method	result	size
risch	\(4 \ln \left (\ln \left (-\ln \relax (x )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}+x^{2} \ln \relax (x )-\ln \relax (3)+\ln \relax (5)\right )\right )\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^2*ln(x)-4*x^2+4)/(x*ln(x^2)-x^3*ln(x)+x*ln(3/5/x))/ln(-ln(x^2)+x^2*ln(x)-ln(3/5/x)),x,method=_RETURN
VERBOSE)

[Out]

4*ln(ln(-ln(x)+1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2+x^2*ln(x)-ln(3)+ln(5)))

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maxima [A]  time = 0.84, size = 19, normalized size = 0.66 \begin {gather*} 4 \, \log \left (\log \left ({\left (x^{2} - 1\right )} \log \relax (x) + \log \relax (5) - \log \relax (3)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^2*log(x)-4*x^2+4)/(x*log(x^2)-x^3*log(x)+x*log(3/5/x))/log(-log(x^2)+x^2*log(x)-log(3/5/x)),x,
algorithm="maxima")

[Out]

4*log(log((x^2 - 1)*log(x) + log(5) - log(3)))

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mupad [B]  time = 1.80, size = 17, normalized size = 0.59 \begin {gather*} 4\,\ln \left (\ln \left (x^2\,\ln \relax (x)-\ln \left (\frac {3\,x}{5}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^2*log(x) + 4*x^2 - 4)/(log(x^2*log(x) - log(3/(5*x)) - log(x^2))*(x*log(x^2) - x^3*log(x) + x*log(3/
(5*x)))),x)

[Out]

4*log(log(x^2*log(x) - log((3*x)/5)))

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sympy [A]  time = 0.68, size = 19, normalized size = 0.66 \begin {gather*} 4 \log {\left (\log {\left (x^{2} \log {\relax (x )} - \log {\relax (x )} - \log {\left (\frac {3}{5} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**2*ln(x)-4*x**2+4)/(x*ln(x**2)-x**3*ln(x)+x*ln(3/5/x))/ln(-ln(x**2)+x**2*ln(x)-ln(3/5/x)),x)

[Out]

4*log(log(x**2*log(x) - log(x) - log(3/5)))

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