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 = 0.03, antiderivative size = 44, 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} \begin {gather*} \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}} \end {gather*}
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}-\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*}
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Mathematica [C] time = 0.10, size = 99, normalized size = 2.25 \begin {gather*} \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 )}} \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}{1-2 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.73, size = 26, normalized size = 0.59 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {3} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {3} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 0.91 \begin {gather*} \frac {\sqrt {2}\, \arctan \left (\frac {\left (4 x -\sqrt {6}\right ) \sqrt {2}}{2}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (4 x +\sqrt {6}\right ) \sqrt {2}}{2}\right )}{2} \end {gather*}
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 \begin {gather*} \int \frac {2 \, x^{2} + 1}{4 \, x^{4} - 2 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 21, normalized size = 0.48 \begin {gather*} \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\sqrt {2}\,x\right )+\mathrm {atan}\left (2\,\sqrt {2}\,x^3\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 29, normalized size = 0.66 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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