3.39.43 \(\int \frac {1-4 x-2 x^2}{2 x-4 x^2-x^3+x \log (x)} \, dx\)

Optimal. Leaf size=12 \[ \log (2+x-x (5+x)+\log (x)) \]

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Rubi [F]  time = 0.21, 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 {1-4 x-2 x^2}{2 x-4 x^2-x^3+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 13, normalized size = 1.08 \begin {gather*} \log \left (-2+4 x+x^2-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-2 + 4*x + x^2 - Log[x]]

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fricas [A]  time = 0.57, size = 13, normalized size = 1.08 \begin {gather*} \log \left (-x^{2} - 4 \, x + \log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-x^2 - 4*x + log(x) + 2)

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giac [A]  time = 0.12, size = 13, normalized size = 1.08 \begin {gather*} \log \left (-x^{2} - 4 \, x + \log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-x^2 - 4*x + log(x) + 2)

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maple [A]  time = 0.02, size = 14, normalized size = 1.17

method	result	size
norman	\(\ln \left (x^{2}+4 x -\ln \relax (x )-2\right )\)	\(14\)
risch	\(\ln \left (-x^{2}+\ln \relax (x )-4 x +2\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.37, size = 13, normalized size = 1.08 \begin {gather*} \log \left (-x^{2} - 4 \, x + \log \relax (x) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-x^2 - 4*x + log(x) + 2)

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mupad [B]  time = 2.26, size = 13, normalized size = 1.08 \begin {gather*} \ln \left (4\,x-\ln \relax (x)+x^2-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(4*x - log(x) + x^2 - 2)

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sympy [A]  time = 0.13, size = 12, normalized size = 1.00 \begin {gather*} \log {\left (- x^{2} - 4 x + \log {\relax (x )} + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-x**2 - 4*x + log(x) + 2)

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