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

Optimal result

Integrand size = 7, antiderivative size = 10 \[ \int \frac {1}{4-6 x} \, dx=-\frac {1}{6} \log (2-3 x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {31} \[ \int \frac {1}{4-6 x} \, dx=-\frac {1}{6} \log (2-3 x) \]

[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*} \text {integral}& = -\frac {1}{6} \log (2-3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{4-6 x} \, dx=-\frac {1}{6} \log (4-6 x) \]

[In]

Integrate[(4 - 6*x)^(-1),x]

[Out]

-1/6*Log[4 - 6*x]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70

method	result	size
parallelrisch	\(-\frac {\ln \left (-\frac {2}{3}+x \right )}{6}\)	\(7\)
default	\(-\frac {\ln \left (2-3 x \right )}{6}\)	\(9\)
norman	\(-\frac {\ln \left (-4+6 x \right )}{6}\)	\(9\)
meijerg	\(-\frac {\ln \left (1-\frac {3 x}{2}\right )}{6}\)	\(9\)
risch	\(-\frac {\ln \left (-2+3 x \right )}{6}\)	\(9\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{4-6 x} \, dx=-\frac {1}{6} \, \log \left (3 \, x - 2\right ) \]

[In]

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

[Out]

-1/6*log(3*x - 2)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{4-6 x} \, dx=- \frac {\log {\left (6 x - 4 \right )}}{6} \]

[In]

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

[Out]

-log(6*x - 4)/6

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{4-6 x} \, dx=-\frac {1}{6} \, \log \left (3 \, x - 2\right ) \]

[In]

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

[Out]

-1/6*log(3*x - 2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{4-6 x} \, dx=-\frac {1}{6} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \]

[In]

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

[Out]

-1/6*log(abs(3*x - 2))

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {1}{4-6 x} \, dx=-\frac {\ln \left (x-\frac {2}{3}\right )}{6} \]

[In]

int(-1/(6*x - 4),x)

[Out]

-log(x - 2/3)/6