Optimal. Leaf size=23 \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3+1}{\sqrt{7}}\right )}{3 \sqrt{7}} \]
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Rubi [A] time = 0.0241188, antiderivative size = 23, 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 = {1352, 618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3+1}{\sqrt{7}}\right )}{3 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 1352
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{x^2}{2+x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{2+x+x^2} \, dx,x,x^3\right )\\ &=-\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,1+2 x^3\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{1+2 x^3}{\sqrt{7}}\right )}{3 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0076714, size = 23, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 x^3+1}{\sqrt{7}}\right )}{3 \sqrt{7}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 0.8 \begin{align*}{\frac{2\,\sqrt{7}}{21}\arctan \left ({\frac{ \left ( 2\,{x}^{3}+1 \right ) \sqrt{7}}{7}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55674, size = 24, normalized size = 1.04 \begin{align*} \frac{2}{21} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{3} + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45253, size = 62, normalized size = 2.7 \begin{align*} \frac{2}{21} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{3} + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.11846, size = 27, normalized size = 1.17 \begin{align*} \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{2 \sqrt{7} x^{3}}{7} + \frac{\sqrt{7}}{7} \right )}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20174, size = 24, normalized size = 1.04 \begin{align*} \frac{2}{21} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{3} + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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