Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{5}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{5}-4 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0414338, antiderivative size = 46, 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} \[ \frac{\tan ^{-1}\left (\frac{4 x+\sqrt{5}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt{5}-4 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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{5} x}{2}+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\frac{\sqrt{5} x}{2}+x^2} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{4}-x^2} \, dx,x,-\frac{\sqrt{5}}{2}+2 x\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{4}-x^2} \, dx,x,\frac{\sqrt{5}}{2}+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{5}-4 x}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{5}+4 x}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.278065, size = 101, normalized size = 2.2 \[ \frac{\left (\sqrt{15}-5 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (-1-i \sqrt{15}\right )}}\right )}{\sqrt{30 \left (-1-i \sqrt{15}\right )}}+\frac{\left (\sqrt{15}+5 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{\frac{1}{2} \left (-1+i \sqrt{15}\right )}}\right )}{\sqrt{30 \left (-1+i \sqrt{15}\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 40, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{5} \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{5} \right ) \sqrt{3}}{3}} \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 = 1.39883, size = 109, normalized size = 2.37 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (4 \, x^{3} + x\right )}\right ) + \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.115995, size = 42, normalized size = 0.91 \begin{align*} \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{4 \sqrt{3} x^{3}}{3} + \frac{\sqrt{3} x}{3} \right )}\right )}{6} \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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