Optimal. Leaf size=9 \[ \tan ^{-1}(2 x)-\tan ^{-1}(x) \]
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Rubi [A] time = 0.0092982, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1163, 203} \[ \tan ^{-1}(2 x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1163
Rule 203
Rubi steps
\begin{align*} \int \frac{1-2 x^2}{1+5 x^2+4 x^4} \, dx &=2 \int \frac{1}{1+4 x^2} \, dx-4 \int \frac{1}{4+4 x^2} \, dx\\ &=-\tan ^{-1}(x)+\tan ^{-1}(2 x)\\ \end{align*}
Mathematica [A] time = 0.0077656, size = 12, normalized size = 1.33 \[ \tan ^{-1}\left (\frac{x}{2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 10, normalized size = 1.1 \begin{align*} -\arctan \left ( x \right ) +\arctan \left ( 2\,x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44267, size = 12, normalized size = 1.33 \begin{align*} \arctan \left (2 \, x\right ) - \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31286, size = 47, normalized size = 5.22 \begin{align*} \arctan \left (4 \, x^{3} + 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 = 0.101973, size = 14, normalized size = 1.56 \begin{align*} - \operatorname{atan}{\left (2 x \right )} + \operatorname{atan}{\left (4 x^{3} + 3 x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11698, size = 12, normalized size = 1.33 \begin{align*} \arctan \left (2 \, x\right ) - \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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