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, 31, 29}
\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 &=-\left (-\frac {(-6-9 x) \int \frac {1}{(-6-9 x) x} \, dx}{\sqrt {-4-12 x-9 x^2}}\right )\\ &=-\frac {(3 (-6-9 x)) \int \frac {1}{-6-9 x} \, dx}{2 \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 {(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 = 33, normalized size = 1.22 \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 = 0.64, size = 30, normalized size = 1.11
method | result | size |
meijerg | \(-\frac {i \left (-\ln \left (1+\frac {3 x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )+\ln \left (3\right )\right )}{2}\) | \(21\) |
default | \(-\frac {\left (2+3 x \right ) \left (-\ln \left (x \right )+\ln \left (2+3 x \right )\right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(30\) |
risch | \(\frac {\left (2+3 x \right ) \ln \left (x \right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}-\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 24, normalized size = 0.89 \begin {gather*} -\frac {1}{2} i \, \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 [C] Result contains complex when optimal does not.
time = 2.81, size = 11, normalized size = 0.41 \begin {gather*} \frac {1}{2} i \, \log \left (x + \frac {2}{3}\right ) - \frac {1}{2} i \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- \left (3 x + 2\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.12, size = 31, normalized size = 1.15 \begin {gather*} -\frac {i \, \log \left ({\left | 3 \, x + 2 \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x - 2\right )} + \frac {i \, \log \left ({\left | x \right |}\right )}{2 \, \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 = 1.24, size = 27, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\frac {6\,x+4-\sqrt {-{\left (3\,x+2\right )}^2}\,2{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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