Optimal. Leaf size=44 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1163, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1163
Rubi steps
\begin {align*} \int \frac {1-x^2}{1+4 x^2+x^4} \, dx &=\frac {1}{2} \left (-1-\sqrt {3}\right ) \int \frac {1}{2+\sqrt {3}+x^2} \, dx+\frac {1}{2} \left (-1+\sqrt {3}\right ) \int \frac {1}{2-\sqrt {3}+x^2} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 82, normalized size = 1.86 \begin {gather*} \frac {-\left (\left (\sqrt {3}-3\right ) \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )\right )-\sqrt {2-\sqrt {3}} \left (3+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3}} \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-x^2}{1+4 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.93, size = 31, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{3} + 3 \, x\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \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 = 26, normalized size = 0.59 \begin {gather*} \frac {1}{4} \, \sqrt {2} {\left (\pi \mathrm {sgn}\relax (x) - 2 \, \arctan \left (\frac {\sqrt {2} {\left (x^{2} + 1\right )}}{2 \, x}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 111, normalized size = 2.52 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{\sqrt {6}-\sqrt {2}}-\frac {\arctan \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{\sqrt {6}-\sqrt {2}}-\frac {\sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{\sqrt {6}+\sqrt {2}}-\frac {\arctan \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{\sqrt {6}+\sqrt {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 {x^{2} - 1}{x^{4} + 4 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 31, normalized size = 0.70 \begin {gather*} \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,x^3}{2}+\frac {3\,\sqrt {2}\,x}{2}\right )-\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\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 = 42, normalized size = 0.95 \begin {gather*} - \frac {\sqrt {2} \left (2 \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {2} x^{3}}{2} + \frac {3 \sqrt {2} x}{2} \right )}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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