Optimal. Leaf size=57 \[ -\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (2 x^4-\sqrt {5}+3\right )-\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (2 x^4+\sqrt {5}+3\right )+\log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1357, 705, 29, 632, 31} \[ -\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (2 x^4-\sqrt {5}+3\right )-\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (2 x^4+\sqrt {5}+3\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 632
Rule 705
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+3 x^4+x^8\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \left (1+3 x+x^2\right )} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-3-x}{1+3 x+x^2} \, dx,x,x^4\right )\\ &=\log (x)+\frac {1}{40} \left (-5+3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,x^4\right )-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,x^4\right )\\ &=\log (x)-\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3-\sqrt {5}+2 x^4\right )-\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (3+\sqrt {5}+2 x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.96 \[ \frac {1}{40} \left (-5-3 \sqrt {5}\right ) \log \left (-2 x^4+\sqrt {5}-3\right )+\frac {1}{40} \left (3 \sqrt {5}-5\right ) \log \left (2 x^4+\sqrt {5}+3\right )+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 58, normalized size = 1.02 \[ \frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{8} + 6 \, x^{4} + \sqrt {5} {\left (2 \, x^{4} + 3\right )} + 7}{x^{8} + 3 \, x^{4} + 1}\right ) - \frac {1}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 51, normalized size = 0.89 \[ -\frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} + 3}{2 \, x^{4} + \sqrt {5} + 3}\right ) - \frac {1}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.61 \[ \frac {3 \sqrt {5}\, \arctanh \left (\frac {\left (2 x^{4}+3\right ) \sqrt {5}}{5}\right )}{20}+\ln \relax (x )-\frac {\ln \left (x^{8}+3 x^{4}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 51, normalized size = 0.89 \[ -\frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} + 3}{2 \, x^{4} + \sqrt {5} + 3}\right ) - \frac {1}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 42, normalized size = 0.74 \[ \ln \relax (x)-\ln \left (x^4-\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {3\,\sqrt {5}}{40}+\frac {1}{8}\right )+\ln \left (x^4+\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {3\,\sqrt {5}}{40}-\frac {1}{8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 58, normalized size = 1.02 \[ \log {\relax (x )} + \left (- \frac {3 \sqrt {5}}{40} - \frac {1}{8}\right ) \log {\left (x^{4} - \frac {\sqrt {5}}{2} + \frac {3}{2} \right )} + \left (- \frac {1}{8} + \frac {3 \sqrt {5}}{40}\right ) \log {\left (x^{4} + \frac {\sqrt {5}}{2} + \frac {3}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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