Optimal. Leaf size=27 \[ -\frac {(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt {9 x^2+12 x+4}} \]
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Rubi [B] 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 = {646, 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 646
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 {(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt {(3 x+2)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {4+12 x+9 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 13, normalized size = 0.48 \begin {gather*} -\frac {1}{2} \, \log \left (3 \, x + 2\right ) + \frac {1}{2} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, 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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maple [A] time = 0.06, size = 28, normalized size = 1.04 \begin {gather*} -\frac {\left (3 x +2\right ) \left (-\ln \relax (x )+\ln \left (3 x +2\right )\right )}{2 \sqrt {\left (3 x +2\right )^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.95, 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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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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sympy [A] time = 0.12, size = 12, normalized size = 0.44 \begin {gather*} \frac {\log {\relax (x )}}{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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