Optimal. Leaf size=15 \[ \tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right ) \]
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Rubi [A] time = 0.0390042, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1698, 203} \[ \tan ^{-1}\left (\frac{x}{\sqrt{x^4+x^2+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 1698
Rule 203
Rubi steps
\begin{align*} \int \frac{1-x^2}{\left (1+x^2\right ) \sqrt{1+x^2+x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{1+x^2+x^4}}\right )\\ &=\tan ^{-1}\left (\frac{x}{\sqrt{1+x^2+x^4}}\right )\\ \end{align*}
Mathematica [C] time = 0.129821, size = 94, normalized size = 6.27 \[ -\frac{(-1)^{2/3} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left (\text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+2 \Pi \left (\sqrt [3]{-1};-i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )}{\sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 188, normalized size = 12.5 \begin{align*} -2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{3}},1/2\,\sqrt{-2+2\,i\sqrt{3}} \right ) }{\sqrt{-2+2\,i\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}}+2\,{\frac{\sqrt{1+1/2\,{x}^{2}-i/2{x}^{2}\sqrt{3}}\sqrt{1+1/2\,{x}^{2}+i/2{x}^{2}\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}{\it EllipticPi} \left ( \sqrt{-1/2+i/2\sqrt{3}}x,- \left ( -1/2+i/2\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-1/2-i/2\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 1}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7684, size = 42, normalized size = 2.8 \begin{align*} \arctan \left (\frac{x}{\sqrt{x^{4} + x^{2} + 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{x^{2} \sqrt{x^{4} + x^{2} + 1} + \sqrt{x^{4} + x^{2} + 1}}\, dx - \int - \frac{1}{x^{2} \sqrt{x^{4} + x^{2} + 1} + \sqrt{x^{4} + x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2} - 1}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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