Optimal. Leaf size=14 \[ \frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {28, 21, 203} \begin {gather*} \frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 28
Rule 203
Rubi steps
\begin {align*} \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx &=4 \int \frac {1+2 x^2}{\left (2+4 x^2\right )^2} \, dx\\ &=\int \frac {1}{1+2 x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.69, size = 11, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 11, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x \right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.30, size = 11, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 11, normalized size = 0.79 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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