Optimal. Leaf size=9 \[ \tan ^{-1}(2 x)-\tan ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 203
Rule 1163
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*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.33 \[ \tan ^{-1}\left (\frac {x}{2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 17, normalized size = 1.89 \[ \arctan \left (4 \, x^{3} + 3 \, x\right ) - \arctan \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 9, normalized size = 1.00 \[ \arctan \left (2 \, x\right ) - \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 10, normalized size = 1.11 \[ -\arctan \relax (x )+\arctan \left (2 x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 9, normalized size = 1.00 \[ \arctan \left (2 \, x\right ) - \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 17, normalized size = 1.89 \[ \mathrm {atan}\left (4\,x^3+3\,x\right )-\mathrm {atan}\left (2\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 1.56 \[ - \operatorname {atan}{\left (2 x \right )} + \operatorname {atan}{\left (4 x^{3} + 3 x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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