Optimal. Leaf size=39 \[ -\frac {\tan ^{-1}\left (\frac {2 x^5+1}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {1}{10} \log \left (x^{10}+x^5+1\right )+\log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1357, 705, 29, 634, 618, 204, 628} \[ -\frac {1}{10} \log \left (x^{10}+x^5+1\right )-\frac {\tan ^{-1}\left (\frac {2 x^5+1}{\sqrt {3}}\right )}{5 \sqrt {3}}+\log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 204
Rule 618
Rule 628
Rule 634
Rule 705
Rule 1357
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+x^5+x^{10}\right )} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x \left (1+x+x^2\right )} \, dx,x,x^5\right )\\ &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^5\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {-1-x}{1+x+x^2} \, dx,x,x^5\right )\\ &=\log (x)-\frac {1}{10} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^5\right )-\frac {1}{10} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^5\right )\\ &=\log (x)-\frac {1}{10} \log \left (1+x^5+x^{10}\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^5\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1+2 x^5}{\sqrt {3}}\right )}{5 \sqrt {3}}+\log (x)-\frac {1}{10} \log \left (1+x^5+x^{10}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 197, normalized size = 5.05 \[ -\frac {1}{5} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^7+\text {$\#$1}^5-\text {$\#$1}^4+\text {$\#$1}^3-\text {$\#$1}+1\& ,\frac {4 \text {$\#$1}^7 \log (x-\text {$\#$1})-3 \text {$\#$1}^6 \log (x-\text {$\#$1})-\text {$\#$1}^5 \log (x-\text {$\#$1})+3 \text {$\#$1}^4 \log (x-\text {$\#$1})-\text {$\#$1}^3 \log (x-\text {$\#$1})+2 \text {$\#$1}^2 \log (x-\text {$\#$1})-\text {$\#$1} \log (x-\text {$\#$1})}{8 \text {$\#$1}^7-7 \text {$\#$1}^6+5 \text {$\#$1}^4-4 \text {$\#$1}^3+3 \text {$\#$1}^2-1}\& \right ]-\frac {1}{10} \log \left (x^2+x+1\right )+\log (x)+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{5 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 32, normalized size = 0.82 \[ -\frac {1}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) - \frac {1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 33, normalized size = 0.85 \[ -\frac {1}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) - \frac {1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 66, normalized size = 1.69 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x^{5}}{3}+\frac {\sqrt {3}}{3}\right )}{15}+\ln \relax (x )-\frac {\ln \left (x^{2}+x +1\right )}{10}-\frac {\ln \left (4 x^{8}-4 x^{7}+4 x^{5}-4 x^{4}+4 x^{3}-4 x +4\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.03, size = 36, normalized size = 0.92 \[ -\frac {1}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{5} + 1\right )}\right ) - \frac {1}{10} \, \log \left (x^{10} + x^{5} + 1\right ) + \frac {1}{5} \, \log \left (x^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 34, normalized size = 0.87 \[ \ln \relax (x)-\frac {\ln \left (x^{10}+x^5+1\right )}{10}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^5}{3}+\frac {\sqrt {3}}{3}\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 41, normalized size = 1.05 \[ \log {\relax (x )} - \frac {\log {\left (x^{10} + x^{5} + 1 \right )}}{10} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{5}}{3} + \frac {\sqrt {3}}{3} \right )}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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