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 = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 204
Rule 618
Rule 1161
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*}
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Mathematica [C] time = 0.10, 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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fricas [A] time = 0.39, size = 29, normalized size = 0.60 \[ \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {2}{3} \, \sqrt {6} {\left (x^{3} + x\right )}\right ) + \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{3} \, \sqrt {6} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 45, normalized size = 0.94 \[ \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {4}{3} \, \sqrt {3} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (2 \, x + \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {4}{3} \, \sqrt {3} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (2 \, x - \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 40, normalized size = 0.83 \[ \frac {\sqrt {6}\, \arctan \left (\frac {\left (4 x -\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (4 x +\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, x^{2} + 1}{4 \, x^{4} + 2 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 29, normalized size = 0.60 \[ \frac {\sqrt {6}\,\left (\mathrm {atan}\left (\frac {2\,\sqrt {6}\,x^3}{3}+\frac {2\,\sqrt {6}\,x}{3}\right )+\mathrm {atan}\left (\frac {\sqrt {6}\,x}{3}\right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 42, normalized size = 0.88 \[ \frac {\sqrt {6} \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} \right )} + 2 \operatorname {atan}{\left (\frac {2 \sqrt {6} x^{3}}{3} + \frac {2 \sqrt {6} x}{3} \right )}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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