Optimal. Leaf size=23 \[ \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-2 \tan ^{-1}\left (\frac {x}{2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1130, 203} \[ \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-2 \tan ^{-1}\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1130
Rubi steps
\begin {align*} \int \frac {x^2}{20+9 x^2+x^4} \, dx &=-\left (4 \int \frac {1}{4+x^2} \, dx\right )+5 \int \frac {1}{5+x^2} \, dx\\ &=-2 \tan ^{-1}\left (\frac {x}{2}\right )+\sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )-2 \tan ^{-1}\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 18, normalized size = 0.78 \[ \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - 2 \, \arctan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 18, normalized size = 0.78 \[ \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - 2 \, \arctan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 0.83 \[ -2 \arctan \left (\frac {x}{2}\right )+\sqrt {5}\, \arctan \left (\frac {\sqrt {5}\, x}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 18, normalized size = 0.78 \[ \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - 2 \, \arctan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 18, normalized size = 0.78 \[ \sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,x}{5}\right )-2\,\mathrm {atan}\left (\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 20, normalized size = 0.87 \[ - 2 \operatorname {atan}{\left (\frac {x}{2} \right )} + \sqrt {5} \operatorname {atan}{\left (\frac {\sqrt {5} x}{5} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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