3.272 \(\int \frac{1}{4-6 x} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{6} \log (2-3 x) \]

[Out]

-Log[2 - 3*x]/6

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Rubi [A]  time = 0.0009693, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {31} \[ -\frac{1}{6} \log (2-3 x) \]

Antiderivative was successfully verified.

[In]

Int[(4 - 6*x)^(-1),x]

[Out]

-Log[2 - 3*x]/6

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{4-6 x} \, dx &=-\frac{1}{6} \log (2-3 x)\\ \end{align*}

Mathematica [A]  time = 0.0015938, size = 10, normalized size = 1. \[ -\frac{1}{6} \log (4-6 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 - 6*x)^(-1),x]

[Out]

-Log[4 - 6*x]/6

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Maple [A]  time = 0., size = 9, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ( 4-6\,x \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4-6*x),x)

[Out]

-1/6*ln(4-6*x)

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Maxima [A]  time = 1.02901, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{6} \, \log \left (3 \, x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*x),x, algorithm="maxima")

[Out]

-1/6*log(3*x - 2)

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Fricas [A]  time = 1.4907, size = 26, normalized size = 2.6 \begin{align*} -\frac{1}{6} \, \log \left (3 \, x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*x),x, algorithm="fricas")

[Out]

-1/6*log(3*x - 2)

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Sympy [A]  time = 0.086162, size = 8, normalized size = 0.8 \begin{align*} - \frac{\log{\left (6 x - 4 \right )}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*x),x)

[Out]

-log(6*x - 4)/6

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Giac [A]  time = 1.15047, size = 12, normalized size = 1.2 \begin{align*} -\frac{1}{6} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-6*x),x, algorithm="giac")

[Out]

-1/6*log(abs(3*x - 2))