Optimal. Leaf size=39 \[ \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )-\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 39, 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*} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )-\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1163
Rubi steps
\begin {align*} \int \frac {1-x^2}{1+3 x^2+x^4} \, dx &=\frac {1}{2} \left (-1-\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx+\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx\\ &=-\tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )+\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 0.26 \begin {gather*} \tan ^{-1}\left (\frac {x}{x^2+1}\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-x^2}{1+3 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.89, size = 13, normalized size = 0.33 \begin {gather*} \arctan \left (x^{3} + 2 \, x\right ) - \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 26, normalized size = 0.67 \begin {gather*} \frac {1}{4} \, \pi \mathrm {sgn}\relax (x) - \frac {1}{2} \, \arctan \left (\frac {x^{4} + x^{2} + 1}{2 \, {\left (x^{3} + 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 = 104, normalized size = 2.67 \begin {gather*} \frac {2 \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{2 \sqrt {5}-2}-\frac {2 \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{2 \sqrt {5}-2}-\frac {2 \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{2 \sqrt {5}+2}-\frac {2 \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{2 \sqrt {5}+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} + 3 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 13, normalized size = 0.33 \begin {gather*} \mathrm {atan}\left (x^3+2\,x\right )-\mathrm {atan}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 10, normalized size = 0.26 \begin {gather*} - \operatorname {atan}{\relax (x )} + \operatorname {atan}{\left (x^{3} + 2 x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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