Optimal. Leaf size=14 \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{3} \left (x^2+1\right )\right ) \]
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Rubi [A] time = 0.013418, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {1107, 618, 204} \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{3} \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Rule 1107
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{x}{10+2 x^2+x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{10+2 x+x^2} \, dx,x,x^2\right )\\ &=-\operatorname{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*}
Mathematica [A] time = 0.0043375, size = 14, normalized size = 1. \[ \frac{1}{6} \tan ^{-1}\left (\frac{1}{3} \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{1}{6}\arctan \left ({\frac{{x}^{2}}{3}}+{\frac{1}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41837, size = 14, normalized size = 1. \begin{align*} \frac{1}{6} \, \arctan \left (\frac{1}{3} \, x^{2} + \frac{1}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83923, size = 36, normalized size = 2.57 \begin{align*} \frac{1}{6} \, \arctan \left (\frac{1}{3} \, x^{2} + \frac{1}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.095847, size = 10, normalized size = 0.71 \begin{align*} \frac{\operatorname{atan}{\left (\frac{x^{2}}{3} + \frac{1}{3} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05179, size = 14, normalized size = 1. \begin{align*} \frac{1}{6} \, \arctan \left (\frac{1}{3} \, x^{2} + \frac{1}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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