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 = 0.0022819, 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, Rules used = {36, 29, 31} \[ \frac{\log (x)}{4}-\frac{1}{4} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(4 + 6*x)),x]

[Out]

Log[x]/4 - Log[2 + 3*x]/4

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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}{x (4+6 x)} \, dx &=\frac{1}{4} \int \frac{1}{x} \, dx-\frac{3}{2} \int \frac{1}{4+6 x} \, dx\\ &=\frac{\log (x)}{4}-\frac{1}{4} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0035897, 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 = 0.002, size = 14, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{4}}-{\frac{\ln \left ( 2+3\,x \right ) }{4}} \end{align*}

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 = 1.06633, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{4} \, \log \left (3 \, x + 2\right ) + \frac{1}{4} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(3*x + 2) + 1/4*log(x)

________________________________________________________________________________________

Fricas [A]  time = 1.70136, size = 43, normalized size = 2.53 \begin{align*} -\frac{1}{4} \, \log \left (3 \, x + 2\right ) + \frac{1}{4} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*log(3*x + 2) + 1/4*log(x)

________________________________________________________________________________________

Sympy [A]  time = 0.110987, size = 12, normalized size = 0.71 \begin{align*} \frac{\log{\left (x \right )}}{4} - \frac{\log{\left (x + \frac{2}{3} \right )}}{4} \end{align*}

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 [A]  time = 1.24486, size = 20, normalized size = 1.18 \begin{align*} -\frac{1}{4} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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