Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0090298, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1166, 207} \[ -\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \frac{-5+2 x^2}{6-5 x^2+x^4} \, dx &=\int \frac{1}{-3+x^2} \, dx+\int \frac{1}{-2+x^2} \, dx\\ &=-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [B] time = 0.0178556, size = 69, normalized size = 2.23 \[ \frac{1}{12} \left (3 \sqrt{2} \log \left (\sqrt{2}-x\right )+2 \sqrt{3} \log \left (\sqrt{3}-x\right )-3 \sqrt{2} \log \left (x+\sqrt{2}\right )-2 \sqrt{3} \log \left (x+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 26, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{x\sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42559, size = 58, normalized size = 1.87 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{x - \sqrt{3}}{x + \sqrt{3}}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\frac{x - \sqrt{2}}{x + \sqrt{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12263, size = 142, normalized size = 4.58 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{x^{2} - 2 \, \sqrt{2} x + 2}{x^{2} - 2}\right ) + \frac{1}{6} \, \sqrt{3} \log \left (\frac{x^{2} - 2 \, \sqrt{3} x + 3}{x^{2} - 3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.476763, size = 60, normalized size = 1.94 \begin{align*} \frac{\sqrt{2} \log{\left (x - \sqrt{2} \right )}}{4} - \frac{\sqrt{2} \log{\left (x + \sqrt{2} \right )}}{4} + \frac{\sqrt{3} \log{\left (x - \sqrt{3} \right )}}{6} - \frac{\sqrt{3} \log{\left (x + \sqrt{3} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06288, size = 80, normalized size = 2.58 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} \right |}}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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