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

Optimal. Leaf size=17 \[ \log \left (x \left (4-x-\frac {\log (4 x)}{x}\right )\right ) \]

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Rubi [F]  time = 0.17, 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}{-4 x^2+x^3+x \log (4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 11, normalized size = 0.65 \begin {gather*} \log ((-4+x) x+\log (4 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[(-4 + x)*x + Log[4*x]]

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fricas [A]  time = 0.74, size = 12, normalized size = 0.71 \begin {gather*} \log \left (x^{2} - 4 \, x + \log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 4*x + log(4*x))

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giac [A]  time = 0.19, size = 12, normalized size = 0.71 \begin {gather*} \log \left (x^{2} - 4 \, x + \log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 4*x + log(4*x))

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maple [A]  time = 0.03, size = 13, normalized size = 0.76

method	result	size
norman	\(\ln \left (x^{2}+\ln \left (4 x \right )-4 x \right )\)	\(13\)
risch	\(\ln \left (x^{2}+\ln \left (4 x \right )-4 x \right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.57, size = 14, normalized size = 0.82 \begin {gather*} \log \left (x^{2} - 4 \, x + 2 \, \log \relax (2) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 4*x + 2*log(2) + log(x))

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mupad [B]  time = 2.15, size = 12, normalized size = 0.71 \begin {gather*} \ln \left (\ln \left (4\,x\right )-4\,x+x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(4*x) - 4*x + x^2)

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sympy [A]  time = 0.14, size = 12, normalized size = 0.71 \begin {gather*} \log {\left (x^{2} - 4 x + \log {\left (4 x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 - 4*x + log(4*x))

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