Optimal. Leaf size=21 \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]
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Rubi [A] time = 0.0141699, antiderivative size = 21, normalized size of antiderivative = 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 = {1357, 632, 31} \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]
Antiderivative was successfully verified.
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Rule 1357
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^5}{3+4 x^3+x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{3+4 x+x^2} \, dx,x,x^3\right )\\ &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^3\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{3+x} \, dx,x,x^3\right )\\ &=-\frac{1}{6} \log \left (1+x^3\right )+\frac{1}{2} \log \left (3+x^3\right )\\ \end{align*}
Mathematica [A] time = 0.0041518, size = 21, normalized size = 1. \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}+{\frac{\ln \left ({x}^{3}+3 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30684, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{2} \, \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 [A] time = 1.48159, size = 50, normalized size = 2.38 \begin{align*} \frac{1}{2} \, \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 = 0.116274, size = 15, normalized size = 0.71 \begin{align*} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09246, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{2} \, \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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