Optimal. Leaf size=27 \[ -\frac {(2+3 x) \tanh ^{-1}(1+3 x)}{\sqrt {4+12 x+9 x^2}} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {660, 36, 29, 31}
\begin {gather*} \frac {(3 x+2) \log (x)}{2 \sqrt {9 x^2+12 x+4}}-\frac {(3 x+2) \log (3 x+2)}{2 \sqrt {9 x^2+12 x+4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {4+12 x+9 x^2}} \, dx &=\frac {(6+9 x) \int \frac {1}{x (6+9 x)} \, dx}{\sqrt {4+12 x+9 x^2}}\\ &=\frac {(6+9 x) \int \frac {1}{x} \, dx}{6 \sqrt {4+12 x+9 x^2}}-\frac {(3 (6+9 x)) \int \frac {1}{6+9 x} \, dx}{2 \sqrt {4+12 x+9 x^2}}\\ &=\frac {(2+3 x) \log (x)}{2 \sqrt {4+12 x+9 x^2}}-\frac {(2+3 x) \log (2+3 x)}{2 \sqrt {4+12 x+9 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.15 \begin {gather*} \frac {(2+3 x) (\log (x)-\log (2+3 x))}{2 \sqrt {(2+3 x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 28, normalized size = 1.04
method | result | size |
default | \(\frac {\left (2+3 x \right ) \left (\ln \left (x \right )-\ln \left (2+3 x \right )\right )}{2 \sqrt {\left (2+3 x \right )^{2}}}\) | \(28\) |
risch | \(\frac {\sqrt {\left (2+3 x \right )^{2}}\, \ln \left (x \right )}{6 x +4}-\frac {\sqrt {\left (2+3 x \right )^{2}}\, \ln \left (2+3 x \right )}{2 \left (2+3 x \right )}\) | \(46\) |
meijerg | \(\frac {-\ln \left (1+\frac {3 x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )+\ln \left (3\right )}{\sqrt {\left (2+3 x \right )^{2}}}+\frac {3 x \left (-\ln \left (1+\frac {3 x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )+\ln \left (3\right )\right )}{2 \sqrt {\left (2+3 x \right )^{2}}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 24, normalized size = 0.89 \begin {gather*} -\frac {1}{2} \, \left (-1\right )^{12 \, x + 8} \log \left (\frac {12 \, x}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.32, size = 13, normalized size = 0.48 \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 = 0.44 \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 = 1.22, size = 21, normalized size = 0.78 \begin {gather*} -\frac {1}{2} \, {\left (\log \left ({\left | 3 \, x + 2 \right |}\right ) - \log \left ({\left | x \right |}\right )\right )} \mathrm {sgn}\left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\ln \left (\frac {6\,x+2\,\sqrt {{\left (3\,x+2\right )}^2}+4}{x}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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