Optimal. Leaf size=23 \[ \tan ^{-1}\left (4 x+\sqrt{7}\right )-\tan ^{-1}\left (\sqrt{7}-4 x\right ) \]
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Rubi [A] time = 0.0262838, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1161, 618, 204} \[ \tan ^{-1}\left (4 x+\sqrt{7}\right )-\tan ^{-1}\left (\sqrt{7}-4 x\right ) \]
Antiderivative was successfully verified.
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Rule 1161
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1+2 x^2}{1-3 x^2+4 x^4} \, dx &=\frac{1}{4} \int \frac{1}{\frac{1}{2}-\frac{\sqrt{7} x}{2}+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\frac{\sqrt{7} x}{2}+x^2} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{4}-x^2} \, dx,x,-\frac{\sqrt{7}}{2}+2 x\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{4}-x^2} \, dx,x,\frac{\sqrt{7}}{2}+2 x\right )\\ &=-\tan ^{-1}\left (\sqrt{7}-4 x\right )+\tan ^{-1}\left (\sqrt{7}+4 x\right )\\ \end{align*}
Mathematica [A] time = 0.0070199, size = 14, normalized size = 0.61 \[ -\tan ^{-1}\left (\frac{x}{2 x^2-1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 20, normalized size = 0.9 \begin{align*} \arctan \left ( 4\,x-\sqrt{7} \right ) +\arctan \left ( 4\,x+\sqrt{7} \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{2 \, x^{2} + 1}{4 \, x^{4} - 3 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33888, size = 45, normalized size = 1.96 \begin{align*} \arctan \left (4 \, x^{3} - x\right ) + \arctan \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.099576, size = 12, normalized size = 0.52 \begin{align*} \operatorname{atan}{\left (2 x \right )} + \operatorname{atan}{\left (4 x^{3} - x \right )} \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{2 \, x^{2} + 1}{4 \, x^{4} - 3 \, x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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