Optimal. Leaf size=25 \[ \frac {1}{3} \tanh ^{-1}\left (\frac {2+3 x}{\sqrt {-5+12 x+9 x^2}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 25, 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 = {635, 212}
\begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {3 x+2}{\sqrt {9 x^2+12 x-5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-5+12 x+9 x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {12+18 x}{\sqrt {-5+12 x+9 x^2}}\right )\\ &=\frac {1}{3} \tanh ^{-1}\left (\frac {2+3 x}{\sqrt {-5+12 x+9 x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 24, normalized size = 0.96 \begin {gather*} -\frac {1}{3} \log \left (-2-3 x+\sqrt {-5+12 x+9 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 30, normalized size = 1.20
method | result | size |
trager | \(-\frac {\ln \left (-2-3 x +\sqrt {9 x^{2}+12 x -5}\right )}{3}\) | \(21\) |
default | \(\frac {\ln \left (\frac {\left (9 x +6\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+12 x -5}\right ) \sqrt {9}}{9}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 5.38, size = 22, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, \log \left (18 \, x + 6 \, \sqrt {9 \, x^{2} + 12 \, x - 5} + 12\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 20, normalized size = 0.80 \begin {gather*} -\frac {1}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 12 \, x - 5} - 2\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 {9 x^{2} + 12 x - 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 41, normalized size = 1.64 \begin {gather*} \frac {1}{6} \, \sqrt {9 \, x^{2} + 12 \, x - 5} {\left (3 \, x + 2\right )} + \frac {3}{2} \, \log \left ({\left | -3 \, x + \sqrt {9 \, x^{2} + 12 \, x - 5} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 20, normalized size = 0.80 \begin {gather*} \frac {\ln \left (3\,x+\sqrt {9\,x^2+12\,x-5}+2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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