Optimal. Leaf size=39 \[ -\frac{1}{10} \log \left (x^{10}+x^5+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )}{5 \sqrt{3}}+\log (x) \]
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Rubi [A] time = 0.0356537, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {1594, 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 1594
Rule 1357
Rule 705
Rule 29
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x+x^6+x^{11}} \, dx &=\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*}
Mathematica [C] time = 0.0188937, 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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Maple [B] time = 0.02, size = 66, normalized size = 1.7 \begin{align*} -{\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}}-{\frac{\sqrt{3}}{15}\arctan \left ({\frac{2\,\sqrt{3}{x}^{5}}{3}}+{\frac{\sqrt{3}}{3}} \right ) }+\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{15} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{5} \, \int \frac{4 \, x^{7} - 3 \, x^{6} - x^{5} + 3 \, x^{4} - x^{3} + 2 \, x^{2} - x}{x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1}\,{d x} - \frac{1}{10} \, \log \left (x^{2} + x + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6487, size = 112, normalized size = 2.87 \begin{align*} -\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 (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.166106, size = 41, normalized size = 1.05 \begin{align*} \log{\left (x \right )} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10021, size = 45, normalized size = 1.15 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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