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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{3} \log (3 x+2) \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log (2+3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(-1),x]

[Out]

Log[2 + 3*x]/3

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Maple [A]
time = 0.06, size = 9, normalized size = 0.90

method	result	size
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	\(\frac {\ln \left (2+3 x \right )}{3}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.92, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, \log \left (3 \, x + 2\right ) \end {gather*}

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)

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Fricas [A]
time = 0.93, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, \log \left (3 \, x + 2\right ) \end {gather*}

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)

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Sympy [A]
time = 0.01, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (3 x + 2 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x + 2)/3

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Giac [A]
time = 0.52, size = 9, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.07, size = 6, normalized size = 0.60 \begin {gather*} \frac {\ln \left (x+\frac {2}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x + 2),x)

[Out]

log(x + 2/3)/3

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