Optimal. Leaf size=39 \[ \tan ^{-1}\left (\frac{1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
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Rubi [A] time = 0.025965, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {2095} \[ \tan ^{-1}\left (\frac{1}{2} x \left (x^4-3 x^2+1\right )\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
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Rule 2095
Rubi steps
\begin{align*} \int \frac{6-3 x^2+x^4}{4+5 x^2-5 x^4+x^6} \, dx &=-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (\sqrt{3}+2 x\right )+\tan ^{-1}\left (\frac{1}{2} x \left (1-3 x^2+x^4\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0107197, size = 41, normalized size = 1.05 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2-3\right )}{x^2-2}\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x \left (x^2-3\right )}{2-x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 23, normalized size = 0.6 \begin{align*} \arctan \left ({\frac{{x}^{5}}{2}}-{\frac{3\,{x}^{3}}{2}}+{\frac{x}{2}} \right ) +\arctan \left ({x}^{3} \right ) +\arctan \left ( x \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^{4} - 3 \, x^{2} + 6}{x^{6} - 5 \, x^{4} + 5 \, x^{2} + 4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06022, size = 82, normalized size = 2.1 \begin{align*} \arctan \left (\frac{1}{2} \, x^{5} - \frac{3}{2} \, x^{3} + \frac{1}{2} \, x\right ) + \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.119388, size = 24, normalized size = 0.62 \begin{align*} \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} \right )} + \operatorname{atan}{\left (\frac{x^{5}}{2} - \frac{3 x^{3}}{2} + \frac{x}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11341, size = 30, normalized size = 0.77 \begin{align*} \arctan \left (\frac{1}{2} \, x^{5} - \frac{3}{2} \, x^{3} + \frac{1}{2} \, x\right ) + \arctan \left (x^{3}\right ) + \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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