3.256 \(\int \frac{1}{x (4+6 x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\log (x)}{4}-\frac{1}{4} \log (3 x+2) \]

[Out]

Log[x]/4 - Log[2 + 3*x]/4

_______________________________________________________________________________________

Rubi [A]  time = 0.010587, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\log (x)}{4}-\frac{1}{4} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(4 + 6*x)),x]

[Out]

Log[x]/4 - Log[2 + 3*x]/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.24698, size = 12, normalized size = 0.71 \[ \frac{\log{\left (x \right )}}{4} - \frac{\log{\left (3 x + 2 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(4+6*x),x)

[Out]

log(x)/4 - log(3*x + 2)/4

_______________________________________________________________________________________

Mathematica [A]  time = 0.00380492, size = 17, normalized size = 1. \[ \frac{\log (x)}{4}-\frac{1}{4} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(4 + 6*x)),x]

[Out]

Log[x]/4 - Log[2 + 3*x]/4

_______________________________________________________________________________________

Maple [A]  time = 0., size = 14, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{4}}-{\frac{\ln \left ( 2+3\,x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(4+6*x),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34242, size = 18, normalized size = 1.06 \[ -\frac{1}{4} \, \log \left (3 \, x + 2\right ) + \frac{1}{4} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((3*x + 2)*x),x, algorithm="maxima")

[Out]

-1/4*log(3*x + 2) + 1/4*log(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.205888, size = 18, normalized size = 1.06 \[ -\frac{1}{4} \, \log \left (3 \, x + 2\right ) + \frac{1}{4} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((3*x + 2)*x),x, algorithm="fricas")

[Out]

-1/4*log(3*x + 2) + 1/4*log(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.187333, size = 12, normalized size = 0.71 \[ \frac{\log{\left (x \right )}}{4} - \frac{\log{\left (x + \frac{2}{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(4+6*x),x)

[Out]

log(x)/4 - log(x + 2/3)/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.202029, size = 20, normalized size = 1.18 \[ -\frac{1}{4} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((3*x + 2)*x),x, algorithm="giac")

[Out]

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