Optimal. Leaf size=44 \[ \frac{\tan ^{-1}\left (2 \sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0336888, antiderivative size = 44, 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} \[ \frac{\tan ^{-1}\left (2 \sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}} \]
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}-\sqrt{\frac{3}{2}} x+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\sqrt{\frac{3}{2}} x+x^2} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}-x^2} \, dx,x,-\sqrt{\frac{3}{2}}+2 x\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}-x^2} \, dx,x,\sqrt{\frac{3}{2}}+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{3}+2 \sqrt{2} x\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.103253, size = 99, normalized size = 2.25 \[ \frac{\left (\sqrt{3}-3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{-1-i \sqrt{3}}}\right )}{2 \sqrt{3 \left (-1-i \sqrt{3}\right )}}+\frac{\left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{-1+i \sqrt{3}}}\right )}{2 \sqrt{3 \left (-1+i \sqrt{3}\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 40, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{6} \right ) \sqrt{2}}{2}} \right ) }+{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{6} \right ) \sqrt{2}}{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} - 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 = 1.32637, size = 90, normalized size = 2.05 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (2 \, \sqrt{2} x^{3}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.107194, size = 29, normalized size = 0.66 \begin{align*} \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\sqrt{2} x \right )} + 2 \operatorname{atan}{\left (2 \sqrt{2} x^{3} \right )}\right )}{4} \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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