Optimal. Leaf size=16 \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]
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Rubi [A] time = 0.0029153, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {26, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}} \]
Antiderivative was successfully verified.
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Rule 26
Rule 206
Rubi steps
\begin{align*} \int \frac{2+3 x^2}{4-9 x^4} \, dx &=\int \frac{1}{2-3 x^2} \, dx\\ &=\frac{\tanh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{\sqrt{6}}\\ \end{align*}
Mathematica [A] time = 0.0140431, size = 32, normalized size = 2. \[ \frac{\log \left (3 x+\sqrt{6}\right )-\log \left (\sqrt{6}-3 x\right )}{2 \sqrt{6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 13, normalized size = 0.8 \begin{align*}{\frac{\sqrt{6}}{6}{\it Artanh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.49265, size = 34, normalized size = 2.12 \begin{align*} -\frac{1}{12} \, \sqrt{6} \log \left (\frac{3 \, x - \sqrt{6}}{3 \, x + \sqrt{6}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.40545, size = 77, normalized size = 4.81 \begin{align*} \frac{1}{12} \, \sqrt{6} \log \left (\frac{3 \, x^{2} + 2 \, \sqrt{6} x + 2}{3 \, x^{2} - 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.091549, size = 32, normalized size = 2. \begin{align*} - \frac{\sqrt{6} \log{\left (x - \frac{\sqrt{6}}{3} \right )}}{12} + \frac{\sqrt{6} \log{\left (x + \frac{\sqrt{6}}{3} \right )}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15612, size = 39, normalized size = 2.44 \begin{align*} \frac{1}{12} \, \sqrt{6} \log \left ({\left | x + \frac{1}{3} \, \sqrt{6} \right |}\right ) - \frac{1}{12} \, \sqrt{6} \log \left ({\left | x - \frac{1}{3} \, \sqrt{6} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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