Optimal. Leaf size=34 \[ -\frac {1}{9 x^3}+\frac {1}{6} \log \left (x^3+1\right )-\frac {1}{54} \log \left (x^3+3\right )-\frac {4 \log (x)}{9} \]
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Rubi [A] time = 0.03, antiderivative size = 34, 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, 709, 800} \[ -\frac {1}{9 x^3}+\frac {1}{6} \log \left (x^3+1\right )-\frac {1}{54} \log \left (x^3+3\right )-\frac {4 \log (x)}{9} \]
Antiderivative was successfully verified.
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Rule 709
Rule 800
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (3+4 x^3+x^6\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (3+4 x+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac {1}{9 x^3}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {-4-x}{x \left (3+4 x+x^2\right )} \, dx,x,x^3\right )\\ &=-\frac {1}{9 x^3}+\frac {1}{9} \operatorname {Subst}\left (\int \left (-\frac {4}{3 x}+\frac {3}{2 (1+x)}-\frac {1}{6 (3+x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{9 x^3}-\frac {4 \log (x)}{9}+\frac {1}{6} \log \left (1+x^3\right )-\frac {1}{54} \log \left (3+x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ -\frac {1}{9 x^3}+\frac {1}{6} \log \left (x^3+1\right )-\frac {1}{54} \log \left (x^3+3\right )-\frac {4 \log (x)}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 35, normalized size = 1.03 \[ -\frac {x^{3} \log \left (x^{3} + 3\right ) - 9 \, x^{3} \log \left (x^{3} + 1\right ) + 24 \, x^{3} \log \relax (x) + 6}{54 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 36, normalized size = 1.06 \[ \frac {4 \, x^{3} - 3}{27 \, x^{3}} - \frac {1}{54} \, \log \left ({\left | x^{3} + 3 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x^{3} + 1 \right |}\right ) - \frac {4}{9} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 1.06 \[ -\frac {4 \ln \relax (x )}{9}+\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x^{3}+3\right )}{54}+\frac {\ln \left (x^{2}-x +1\right )}{6}-\frac {1}{9 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 28, normalized size = 0.82 \[ -\frac {1}{9 \, x^{3}} - \frac {1}{54} \, \log \left (x^{3} + 3\right ) + \frac {1}{6} \, \log \left (x^{3} + 1\right ) - \frac {4}{27} \, \log \left (x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 26, normalized size = 0.76 \[ \frac {\ln \left (x^3+1\right )}{6}-\frac {\ln \left (x^3+3\right )}{54}-\frac {4\,\ln \relax (x)}{9}-\frac {1}{9\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 29, normalized size = 0.85 \[ - \frac {4 \log {\relax (x )}}{9} + \frac {\log {\left (x^{3} + 1 \right )}}{6} - \frac {\log {\left (x^{3} + 3 \right )}}{54} - \frac {1}{9 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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