Optimal. Leaf size=48 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{3}}\right )}{\sqrt{6}}-\frac{\tan ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{3}}\right )}{\sqrt{6}} \]
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Rubi [A] time = 0.0395263, 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, 204} \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{3}}\right )}{\sqrt{6}}-\frac{\tan ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{3}}\right )}{\sqrt{6}} \]
Antiderivative was successfully verified.
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Rule 1161
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1+2 x^2}{1+2 x^2+4 x^4} \, dx &=\frac{1}{4} \int \frac{1}{\frac{1}{2}-\frac{x}{\sqrt{2}}+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\frac{x}{\sqrt{2}}+x^2} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}-x^2} \, dx,x,-\frac{1}{\sqrt{2}}+2 x\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}-x^2} \, dx,x,\frac{1}{\sqrt{2}}+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{3}}\right )}{\sqrt{6}}+\frac{\tan ^{-1}\left (\frac{1+2 \sqrt{2} x}{\sqrt{3}}\right )}{\sqrt{6}}\\ \end{align*}
Mathematica [C] time = 0.104063, size = 99, normalized size = 2.06 \[ \frac{\left (\sqrt{3}-i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{1-i \sqrt{3}}}\right )}{2 \sqrt{3 \left (1-i \sqrt{3}\right )}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{1+i \sqrt{3}}}\right )}{2 \sqrt{3 \left (1+i \sqrt{3}\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 40, normalized size = 0.8 \begin{align*}{\frac{\sqrt{6}}{6}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{2} \right ) \sqrt{6}}{6}} \right ) }+{\frac{\sqrt{6}}{6}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{2} \right ) \sqrt{6}}{6}} \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} + 2 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24012, size = 107, normalized size = 2.23 \begin{align*} \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{2}{3} \, \sqrt{6}{\left (x^{3} + x\right )}\right ) + \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{3} \, \sqrt{6} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111128, size = 42, normalized size = 0.88 \begin{align*} \frac{\sqrt{6} \left (2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{2 \sqrt{6} x^{3}}{3} + \frac{2 \sqrt{6} x}{3} \right )}\right )}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{2} + 1}{4 \, x^{4} + 2 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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