Optimal. Leaf size=18 \[ x-\frac {8}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} \tan ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1136, 1180,
209} \begin {gather*} -\frac {8}{3} \text {ArcTan}\left (\frac {x}{2}\right )+\frac {\text {ArcTan}(x)}{3}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1136
Rule 1180
Rubi steps
\begin {align*} \int \frac {x^4}{4+5 x^2+x^4} \, dx &=x-\int \frac {4+5 x^2}{4+5 x^2+x^4} \, dx\\ &=x+\frac {1}{3} \int \frac {1}{1+x^2} \, dx-\frac {16}{3} \int \frac {1}{4+x^2} \, dx\\ &=x-\frac {8}{3} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} x+\frac {8}{3} \tan ^{-1}\left (\frac {2}{x}\right )+\frac {1}{3} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 13, normalized size = 0.72
method | result | size |
default | \(x -\frac {8 \arctan \left (\frac {x}{2}\right )}{3}+\frac {\arctan \left (x \right )}{3}\) | \(13\) |
risch | \(x -\frac {8 \arctan \left (\frac {x}{2}\right )}{3}+\frac {\arctan \left (x \right )}{3}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.57, size = 12, normalized size = 0.67 \begin {gather*} x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.07, size = 12, normalized size = 0.67 \begin {gather*} x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.78 \begin {gather*} x - \frac {8 \operatorname {atan}{\left (\frac {x}{2} \right )}}{3} + \frac {\operatorname {atan}{\left (x \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 12, normalized size = 0.67 \begin {gather*} x - \frac {8}{3} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{3} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 12, normalized size = 0.67 \begin {gather*} x-\frac {8\,\mathrm {atan}\left (\frac {x}{2}\right )}{3}+\frac {\mathrm {atan}\left (x\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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