3.457 \(\int \frac{1}{2+3 x} \, dx\)

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

[Out]

Log[2 + 3*x]/3

________________________________________________________________________________________

Rubi [A]  time = 0.0009813, 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} \[ \frac{1}{3} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(-1),x]

[Out]

Log[2 + 3*x]/3

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}{2+3 x} \, dx &=\frac{1}{3} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0008833, size = 10, normalized size = 1. \[ \frac{1}{3} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(-1),x]

[Out]

Log[2 + 3*x]/3

________________________________________________________________________________________

Maple [A]  time = 0., size = 9, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 2+3\,x \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*x),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.13284, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{3} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.19238, size = 24, normalized size = 2.4 \begin{align*} \frac{1}{3} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(3*x + 2)

________________________________________________________________________________________

Sympy [A]  time = 0.052239, size = 7, normalized size = 0.7 \begin{align*} \frac{\log{\left (3 x + 2 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x),x)

[Out]

log(3*x + 2)/3

________________________________________________________________________________________

Giac [A]  time = 1.23442, size = 12, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(abs(3*x + 2))