3.3.72 \(\int \frac {1}{4-6 x} \, dx\) [272]

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

[Out]

-1/6*ln(2-3*x)

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{6} \log (2-3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 6*x)^(-1),x]

[Out]

-1/6*Log[2 - 3*x]

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*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{6} \log (4-6 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - 6*x)^(-1),x]

[Out]

-1/6*Log[4 - 6*x]

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Mathics [A]
time = 1.57, size = 8, normalized size = 0.80 \begin {gather*} -\frac {\text {Log}\left [-4+6 x\right ]}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(4 - 6*x),x]')

[Out]

-Log[-4 + 6 x] / 6

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Maple [A]
time = 0.08, size = 9, normalized size = 0.90

method	result	size
default	\(-\frac {\ln \left (4-6 x \right )}{6}\)	\(9\)
norman	\(-\frac {\ln \left (6 x -4\right )}{6}\)	\(9\)
meijerg	\(-\frac {\ln \left (1-\frac {3 x}{2}\right )}{6}\)	\(9\)
risch	\(-\frac {\ln \left (-2+3 x \right )}{6}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4-6*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, \log \left (3 \, x - 2\right ) \end {gather*}

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 = 0.31, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{6} \, \log \left (3 \, x - 2\right ) \end {gather*}

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 = 0.03, size = 8, normalized size = 0.80 \begin {gather*} - \frac {\log {\left (6 x - 4 \right )}}{6} \end {gather*}

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 = 0.00, size = 11, normalized size = 1.10 \begin {gather*} -\frac {\ln \left |3 x-2\right |}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*log(abs(3*x - 2))

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Mupad [B]
time = 0.08, size = 6, normalized size = 0.60 \begin {gather*} -\frac {\ln \left (x-\frac {2}{3}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(6*x - 4),x)

[Out]

-log(x - 2/3)/6

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