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 = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 31
Rule 632
Rule 1357
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*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^3+3\right )-\frac {1}{6} \log \left (x^3+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 17, normalized size = 0.81 \[ \frac {1}{2} \, \log \left (x^{3} + 3\right ) - \frac {1}{6} \, \log \left (x^{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 19, normalized size = 0.90 \[ \frac {1}{2} \, \log \left ({\left | x^{3} + 3 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.86 \[ -\frac {\ln \left (x^{3}+1\right )}{6}+\frac {\ln \left (x^{3}+3\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 17, normalized size = 0.81 \[ \frac {1}{2} \, \log \left (x^{3} + 3\right ) - \frac {1}{6} \, \log \left (x^{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 17, normalized size = 0.81 \[ \frac {\ln \left (x^3+3\right )}{2}-\frac {\ln \left (x^3+1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 15, normalized size = 0.71 \[ - \frac {\log {\left (x^{3} + 1 \right )}}{6} + \frac {\log {\left (x^{3} + 3 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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