3.31.32 \(\int \frac {-8-2 x+8 x^2}{-4 x-6 x^2-4 x^3+x^2 \log (x)} \, dx\)

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

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Rubi [F]  time = 0.23, 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 {-8-2 x+8 x^2}{-4 x-6 x^2-4 x^3+x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 - 2*x + 8*x^2)/(-4*x - 6*x^2 - 4*x^3 + x^2*Log[x]),x]

[Out]

2*Defer[Int][(4 + 6*x + 4*x^2 - x*Log[x])^(-1), x] + 8*Defer[Int][1/(x*(4 + 6*x + 4*x^2 - x*Log[x])), x] - 8*D
efer[Int][x/(4 + 6*x + 4*x^2 - x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2}{4+6 x+4 x^2-x \log (x)}+\frac {8}{x \left (4+6 x+4 x^2-x \log (x)\right )}-\frac {8 x}{4+6 x+4 x^2-x \log (x)}\right ) \, dx\\ &=2 \int \frac {1}{4+6 x+4 x^2-x \log (x)} \, dx+8 \int \frac {1}{x \left (4+6 x+4 x^2-x \log (x)\right )} \, dx-8 \int \frac {x}{4+6 x+4 x^2-x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 23, normalized size = 1.00 \begin {gather*} 2 \log (x)-2 \log \left (4+6 x+4 x^2-x \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - 2*x + 8*x^2)/(-4*x - 6*x^2 - 4*x^3 + x^2*Log[x]),x]

[Out]

2*Log[x] - 2*Log[4 + 6*x + 4*x^2 - x*Log[x]]

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fricas [A]  time = 0.86, size = 23, normalized size = 1.00 \begin {gather*} -2 \, \log \left (-\frac {4 \, x^{2} - x \log \relax (x) + 6 \, x + 4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-2*x-8)/(x^2*log(x)-4*x^3-6*x^2-4*x),x, algorithm="fricas")

[Out]

-2*log(-(4*x^2 - x*log(x) + 6*x + 4)/x)

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giac [A]  time = 0.37, size = 22, normalized size = 0.96 \begin {gather*} -2 \, \log \left (-4 \, x^{2} + x \log \relax (x) - 6 \, x - 4\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-2*x-8)/(x^2*log(x)-4*x^3-6*x^2-4*x),x, algorithm="giac")

[Out]

-2*log(-4*x^2 + x*log(x) - 6*x - 4) + 2*log(x)

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maple [A]  time = 0.02, size = 22, normalized size = 0.96

method	result	size
risch	\(-2 \ln \left (\ln \relax (x )-\frac {2 \left (2 x^{2}+3 x +2\right )}{x}\right )\)	\(22\)
norman	\(2 \ln \relax (x )-2 \ln \left (4 x^{2}-x \ln \relax (x )+6 x +4\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^2-2*x-8)/(x^2*ln(x)-4*x^3-6*x^2-4*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 23, normalized size = 1.00 \begin {gather*} -2 \, \log \left (-\frac {4 \, x^{2} - x \log \relax (x) + 6 \, x + 4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-2*x-8)/(x^2*log(x)-4*x^3-6*x^2-4*x),x, algorithm="maxima")

[Out]

-2*log(-(4*x^2 - x*log(x) + 6*x + 4)/x)

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mupad [B]  time = 2.00, size = 21, normalized size = 0.91 \begin {gather*} 2\,\ln \relax (x)-2\,\ln \left (\frac {3\,x}{2}-\frac {x\,\ln \relax (x)}{4}+x^2+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 8*x^2 + 8)/(4*x - x^2*log(x) + 6*x^2 + 4*x^3),x)

[Out]

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

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sympy [A]  time = 0.16, size = 20, normalized size = 0.87 \begin {gather*} - 2 \log {\left (\log {\relax (x )} + \frac {- 4 x^{2} - 6 x - 4}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**2-2*x-8)/(x**2*ln(x)-4*x**3-6*x**2-4*x),x)

[Out]

-2*log(log(x) + (-4*x**2 - 6*x - 4)/x)

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