Optimal. Leaf size=48 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{5}}\right )}{\sqrt{10}}-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}} \]
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Rubi [A] time = 0.0386751, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{5}}\right )}{\sqrt{10}}-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 1161
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1-6 x^2+4 x^4} \, dx &=-\left (\frac{1}{4} \int \frac{1}{-\frac{1}{2}-\frac{x}{\sqrt{2}}+x^2} \, dx\right )-\frac{1}{4} \int \frac{1}{-\frac{1}{2}+\frac{x}{\sqrt{2}}+x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{5}{2}-x^2} \, dx,x,-\frac{1}{\sqrt{2}}+2 x\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{5}{2}-x^2} \, dx,x,\frac{1}{\sqrt{2}}+2 x\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}}+\frac{\tanh ^{-1}\left (\frac{1+2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0190844, size = 42, normalized size = 0.88 \[ \frac{\log \left (2 x^2+\sqrt{10} x+1\right )-\log \left (-2 x^2+\sqrt{10} x-1\right )}{2 \sqrt{10}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 82, normalized size = 1.7 \begin{align*}{\frac{ \left ( -2+2\,\sqrt{5} \right ) \sqrt{5}}{10\,\sqrt{10}-10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{5} \right ) \sqrt{5}}{10\,\sqrt{10}+10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{2 \, x^{2} - 1}{4 \, x^{4} - 6 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27381, size = 116, normalized size = 2.42 \begin{align*} \frac{1}{20} \, \sqrt{10} \log \left (\frac{4 \, x^{4} + 14 \, x^{2} + 2 \, \sqrt{10}{\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} - 6 \, x^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.10442, size = 46, normalized size = 0.96 \begin{align*} - \frac{\sqrt{10} \log{\left (x^{2} - \frac{\sqrt{10} x}{2} + \frac{1}{2} \right )}}{20} + \frac{\sqrt{10} \log{\left (x^{2} + \frac{\sqrt{10} x}{2} + \frac{1}{2} \right )}}{20} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18294, size = 104, normalized size = 2.17 \begin{align*} \frac{1}{20} \, \sqrt{10} \log \left ({\left | x + \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) + \frac{1}{20} \, \sqrt{10} \log \left ({\left | x + \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) - \frac{1}{20} \, \sqrt{10} \log \left ({\left | x - \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) - \frac{1}{20} \, \sqrt{10} \log \left ({\left | x - \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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