Optimal. Leaf size=10 \[ -\frac {1}{3} \tanh ^{-1}\left (x^3+2\right ) \]
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Rubi [B] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 2.10, 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 31
Rule 616
Rule 1352
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*}
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Mathematica [B] time = 0.00, size = 21, normalized size = 2.10 \[ \frac {1}{6} \log \left (x^3+1\right )-\frac {1}{6} \log \left (x^3+3\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.13, size = 17, normalized size = 1.70 \[ -\frac {1}{6} \, \log \left (x^{3} + 3\right ) + \frac {1}{6} \, \log \left (x^{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 19, normalized size = 1.90 \[ -\frac {1}{6} \, \log \left ({\left | x^{3} + 3 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 18, normalized size = 1.80 \[ \frac {\ln \left (x^{3}+1\right )}{6}-\frac {\ln \left (x^{3}+3\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 17, normalized size = 1.70 \[ -\frac {1}{6} \, \log \left (x^{3} + 3\right ) + \frac {1}{6} \, \log \left (x^{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 16, normalized size = 1.60 \[ \frac {\mathrm {atanh}\left (\frac {9}{2\,\left (8\,x^3+6\right )}+\frac {5}{4}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 15, normalized size = 1.50 \[ \frac {\log {\left (x^{3} + 1 \right )}}{6} - \frac {\log {\left (x^{3} + 3 \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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