Optimal. Leaf size=10 \[ -\frac{1}{3} \tanh ^{-1}\left (x^3+2\right ) \]
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Rubi [B] time = 0.0139787, antiderivative size = 21, normalized size of antiderivative = 2.1, 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 = {1352, 616, 31} \[ \frac{1}{6} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^3+3\right ) \]
Antiderivative was successfully verified.
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Rule 1352
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2}{3+4 x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^3\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,x^3\right )\\ &=\frac{1}{6} \log \left (1+x^3\right )-\frac{1}{6} \log \left (3+x^3\right )\\ \end{align*}
Mathematica [B] time = 0.0035071, size = 21, normalized size = 2.1 \[ \frac{1}{6} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^3+3\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 18, normalized size = 1.8 \begin{align*}{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}-{\frac{\ln \left ({x}^{3}+3 \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.38805, size = 23, normalized size = 2.3 \begin{align*} -\frac{1}{6} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.4608, size = 51, normalized size = 5.1 \begin{align*} -\frac{1}{6} \, \log \left (x^{3} + 3\right ) + \frac{1}{6} \, \log \left (x^{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111578, size = 15, normalized size = 1.5 \begin{align*} \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{\log{\left (x^{3} + 3 \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09131, size = 26, normalized size = 2.6 \begin{align*} -\frac{1}{6} \, \log \left ({\left | x^{3} + 3 \right |}\right ) + \frac{1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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