Optimal. Leaf size=17 \[ \frac {\log (x)}{2}-\frac {1}{2} \log (2+3 x) \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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}
\begin {gather*} \frac {\log (x)}{2}-\frac {1}{2} \log (3 x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {1}{x (2+3 x)} \, dx &=\frac {1}{2} \int \frac {1}{x} \, dx-\frac {3}{2} \int \frac {1}{2+3 x} \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \log (2+3 x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{2}-\frac {1}{2} \log (2+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 14, normalized size = 0.82
method | result | size |
default | \(\frac {\ln \left (x \right )}{2}-\frac {\ln \left (2+3 x \right )}{2}\) | \(14\) |
norman | \(\frac {\ln \left (x \right )}{2}-\frac {\ln \left (2+3 x \right )}{2}\) | \(14\) |
risch | \(\frac {\ln \left (x \right )}{2}-\frac {\ln \left (2+3 x \right )}{2}\) | \(14\) |
meijerg | \(\frac {\ln \left (x \right )}{2}-\frac {\ln \left (2\right )}{2}+\frac {\ln \left (3\right )}{2}-\frac {\ln \left (1+\frac {3 x}{2}\right )}{2}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.83, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 12, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x \right )}}{2} - \frac {\log {\left (x + \frac {2}{3} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 15, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 10, normalized size = 0.59 \begin {gather*} -\frac {\ln \left (\frac {2}{x}+3\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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