Optimal. Leaf size=23 \[ -2 \tan ^{-1}\left (\frac {x}{2}\right )+\sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \]
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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 = {1144, 209}
\begin {gather*} \sqrt {5} \text {ArcTan}\left (\frac {x}{\sqrt {5}}\right )-2 \text {ArcTan}\left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1144
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 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{2}\right )+\sqrt {5} \tan ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 19, normalized size = 0.83
method | result | size |
default | \(-2 \arctan \left (\frac {x}{2}\right )+\arctan \left (\frac {x \sqrt {5}}{5}\right ) \sqrt {5}\) | \(19\) |
risch | \(-2 \arctan \left (\frac {x}{2}\right )+\arctan \left (\frac {x \sqrt {5}}{5}\right ) \sqrt {5}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 18, normalized size = 0.78 \begin {gather*} \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - 2 \, \arctan \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 18, normalized size = 0.78 \begin {gather*} \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - 2 \, \arctan \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 20, normalized size = 0.87 \begin {gather*} - 2 \operatorname {atan}{\left (\frac {x}{2} \right )} + \sqrt {5} \operatorname {atan}{\left (\frac {\sqrt {5} x}{5} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.09, size = 18, normalized size = 0.78 \begin {gather*} \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - 2 \, \arctan \left (\frac {1}{2} \, x\right ) \end {gather*}
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 \begin {gather*} \sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,x}{5}\right )-2\,\mathrm {atan}\left (\frac {x}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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