Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{3}}{\sqrt{5}}\right )}{\sqrt{5}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}-4 x}{\sqrt{5}}\right )}{\sqrt{5}} \]
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Rubi [A] time = 0.0431827, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1161, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{3}}{\sqrt{5}}\right )}{\sqrt{5}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3}-4 x}{\sqrt{5}}\right )}{\sqrt{5}} \]
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+x^2+4 x^4} \, dx &=\frac{1}{4} \int \frac{1}{\frac{1}{2}-\frac{\sqrt{3} x}{2}+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\frac{\sqrt{3} x}{2}+x^2} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{5}{4}-x^2} \, dx,x,-\frac{\sqrt{3}}{2}+2 x\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{5}{4}-x^2} \, dx,x,\frac{\sqrt{3}}{2}+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{3}-4 x}{\sqrt{5}}\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3}+4 x}{\sqrt{5}}\right )}{\sqrt{5}}\\ \end{align*}
Mathematica [C] time = 0.225133, size = 97, normalized size = 2.11 \[ \frac{\left (\sqrt{15}-3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (1-i \sqrt{15}\right )}}\right )}{\sqrt{30-30 i \sqrt{15}}}+\frac{\left (\sqrt{15}+3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (1+i \sqrt{15}\right )}}\right )}{\sqrt{30+30 i \sqrt{15}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 40, normalized size = 0.9 \begin{align*}{\frac{\sqrt{5}}{5}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{3} \right ) \sqrt{5}}{5}} \right ) }+{\frac{\sqrt{5}}{5}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{3} \right ) \sqrt{5}}{5}} \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} + x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34345, size = 112, normalized size = 2.43 \begin{align*} \frac{1}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (4 \, x^{3} + 3 \, x\right )}\right ) + \frac{1}{5} \, \sqrt{5} \arctan \left (\frac{2}{5} \, \sqrt{5} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112677, size = 44, normalized size = 0.96 \begin{align*} \frac{\sqrt{5} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{5} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{4 \sqrt{5} x^{3}}{5} + \frac{3 \sqrt{5} x}{5} \right )}\right )}{10} \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} + x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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