\(\int \frac {1}{2+3 x} \, dx\) [56]

Optimal result

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

[Out]

1/3*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} \[ \int \frac {1}{2+3 x} \, dx=\frac {1}{3} \log (3 x+2) \]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2+3 x} \, dx=\frac {1}{3} \log (2+3 x) \]

[In]

Integrate[(2 + 3*x)^(-1),x]

[Out]

Log[2 + 3*x]/3

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70

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

[In]

int(1/(2+3*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{2+3 x} \, dx=\frac {1}{3} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/3*log(3*x + 2)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {1}{2+3 x} \, dx=\frac {\log {\left (3 x + 2 \right )}}{3} \]

[In]

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

[Out]

log(3*x + 2)/3

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{2+3 x} \, dx=\frac {1}{3} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/3*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {1}{2+3 x} \, dx=\frac {1}{3} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

1/3*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int \frac {1}{2+3 x} \, dx=\frac {\ln \left (x+\frac {2}{3}\right )}{3} \]

[In]

int(1/(3*x + 2),x)

[Out]

log(x + 2/3)/3