Optimal. Leaf size=14 \[ \frac {1}{6} \tan ^{-1}\left (\frac {1}{3} \left (1+x^2\right )\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1121, 632, 210}
\begin {gather*} \frac {1}{6} \text {ArcTan}\left (\frac {1}{3} \left (x^2+1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 1121
Rubi steps
\begin {align*} \int \frac {x}{10+2 x^2+x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{10+2 x+x^2} \, dx,x,x^2\right )\\ &=-\text {Subst}\left (\int \frac {1}{-36-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac {1}{6} \tan ^{-1}\left (\frac {1}{3} \left (1+x^2\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{6} \tan ^{-1}\left (\frac {1}{3} \left (1+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 11, normalized size = 0.79
method | result | size |
default | \(\frac {\arctan \left (\frac {x^{2}}{3}+\frac {1}{3}\right )}{6}\) | \(11\) |
risch | \(\frac {\arctan \left (\frac {x^{2}}{3}+\frac {1}{3}\right )}{6}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {1}{3} \, x^{2} + \frac {1}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {1}{3} \, x^{2} + \frac {1}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 10, normalized size = 0.71 \begin {gather*} \frac {\operatorname {atan}{\left (\frac {x^{2}}{3} + \frac {1}{3} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.16, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{6} \, \arctan \left (\frac {1}{3} \, x^{2} + \frac {1}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 10, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x^2}{3}+\frac {1}{3}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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