Optimal. Leaf size=23 \[ \tan ^{-1}\left (2 x+\sqrt{3}\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right ) \]
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Rubi [A] time = 0.0195514, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1161, 618, 204} \[ \tan ^{-1}\left (2 x+\sqrt{3}\right )-\tan ^{-1}\left (\sqrt{3}-2 x\right ) \]
Antiderivative was successfully verified.
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Rule 1161
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1+x^2}{1-x^2+x^4} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{3} x+x^2} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{3} x+x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 x\right )-\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 x\right )\\ &=-\tan ^{-1}\left (\sqrt{3}-2 x\right )+\tan ^{-1}\left (\sqrt{3}+2 x\right )\\ \end{align*}
Mathematica [A] time = 0.0066184, size = 12, normalized size = 0.52 \[ -\tan ^{-1}\left (\frac{x}{x^2-1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 20, normalized size = 0.9 \begin{align*} \arctan \left ( 2\,x-\sqrt{3} \right ) +\arctan \left ( 2\,x+\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}{x^{4} - x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35492, size = 34, normalized size = 1.48 \begin{align*} \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 = 0.101872, size = 7, normalized size = 0.3 \begin{align*} \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1328, size = 41, normalized size = 1.78 \begin{align*} \frac{1}{4} \, \pi \mathrm{sgn}\left (x\right ) + \frac{1}{2} \, \arctan \left (\frac{x^{4} - 3 \, x^{2} + 1}{2 \,{\left (x^{3} - x\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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