Optimal. Leaf size=29 \[ \frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-4-12 x-9 x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {622, 31}
\begin {gather*} \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-9 x^2-12 x-4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 622
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx &=-\left (-\frac {(-6-9 x) \int \frac {1}{-6-9 x} \, dx}{\sqrt {-4-12 x-9 x^2}}\right )\\ &=\frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-4-12 x-9 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 0.97 \begin {gather*} \frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-(2+3 x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 25, normalized size = 0.86
method | result | size |
meijerg | \(-\frac {i \ln \left (1+\frac {3 x}{2}\right )}{3}\) | \(10\) |
default | \(\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{3 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(25\) |
risch | \(\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{3 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 6, normalized size = 0.21 \begin {gather*} \frac {1}{3} i \, \log \left (x + \frac {2}{3}\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 = 1.39, size = 6, normalized size = 0.21 \begin {gather*} -\frac {1}{3} i \, \log \left (x + \frac {2}{3}\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}{\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 = 0.67, size = 23, normalized size = 0.79 \begin {gather*} \frac {i \, \log \left ({\left (-3 i \, x - 2 i\right )} \mathrm {sgn}\left (-3 \, x - 2\right )\right )}{3 \, \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 = 0.29, size = 15, normalized size = 0.52 \begin {gather*} -\frac {\ln \left (-3\,x-2\right )\,\mathrm {sign}\left (3\,x+2\right )\,1{}\mathrm {i}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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