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

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

[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} \[ \frac {1}{3} \log (3 x+2) \]

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*}

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

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(-1),x]

[Out]

Log[2 + 3*x]/3

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fricas [A]  time = 0.39, size = 8, normalized size = 0.80 \[ \frac {1}{3} \, \log \left (3 \, x + 2\right ) \]

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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giac [A]  time = 0.01, size = 9, normalized size = 0.90 \[ \frac {1}{3} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

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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maple [A]  time = 0.00, size = 9, normalized size = 0.90 \[ \frac {\ln \left (3 x +2\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 8, normalized size = 0.80 \[ \frac {1}{3} \, \log \left (3 \, x + 2\right ) \]

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

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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sympy [A]  time = 0.06, size = 7, normalized size = 0.70 \[ \frac {\log {\left (3 x + 2 \right )}}{3} \]

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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