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

Optimal. Leaf size=23 \[ 2 \left (2+3 e^9-x \log \left (4-\log \left (-4 x^2\right )\right )\right ) \]

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Rubi [F]  time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-4+\left (8-2 \log \left (-4 x^2\right )\right ) \log \left (4-\log \left (-4 x^2\right )\right )}{-4+\log \left (-4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((E^2*x*ExpIntegralEi[(-4 + Log[-4*x^2])/2])/Sqrt[-x^2]) - 2*Defer[Int][Log[4 - Log[-4*x^2]], x]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 14, normalized size = 0.61 \begin {gather*} -2 x \log \left (4-\log \left (-4 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*x*Log[4 - Log[-4*x^2]]

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fricas [A]  time = 0.76, size = 14, normalized size = 0.61 \begin {gather*} -2 \, x \log \left (-\log \left (-4 \, x^{2}\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(-log(-4*x^2) + 4)

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giac [A]  time = 0.29, size = 14, normalized size = 0.61 \begin {gather*} -2 \, x \log \left (-\log \left (-4 \, x^{2}\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(-log(-4*x^2) + 4)

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maple [A]  time = 0.05, size = 15, normalized size = 0.65

method	result	size
norman	\(-2 x \ln \left (-\ln \left (-4 x^{2}\right )+4\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*ln(-ln(-4*x^2)+4)

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maxima [C]  time = 0.59, size = 17, normalized size = 0.74 \begin {gather*} -2 \, x \log \left (-i \, \pi - 2 \, \log \relax (2) - 2 \, \log \relax (x) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(-I*pi - 2*log(2) - 2*log(x) + 4)

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mupad [B]  time = 3.99, size = 14, normalized size = 0.61 \begin {gather*} -2\,x\,\ln \left (4-\ln \left (-4\,x^2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(4 - log(-4*x^2))

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sympy [A]  time = 0.32, size = 15, normalized size = 0.65 \begin {gather*} - 2 x \log {\left (4 - \log {\left (- 4 x^{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(4 - log(-4*x**2))

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