Optimal. Leaf size=85 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{x}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
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Rubi [A] time = 0.249722, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {325, 295, 634, 618, 204, 628, 203} \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{x}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
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Rule 325
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (1+x^6\right )} \, dx &=-\frac{1}{x}-\int \frac{x^4}{1+x^6} \, dx\\ &=-\frac{1}{x}-\frac{1}{3} \int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx-\frac{1}{3} \int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx-\frac{1}{3} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{1}{x}-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{12} \int \frac{1}{1-\sqrt{3} x+x^2} \, dx-\frac{1}{12} \int \frac{1}{1+\sqrt{3} x+x^2} \, dx-\frac{\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx}{4 \sqrt{3}}+\frac{\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx}{4 \sqrt{3}}\\ &=-\frac{1}{x}-\frac{1}{3} \tan ^{-1}(x)-\frac{\log \left (1-\sqrt{3} x+x^2\right )}{4 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} x+x^2\right )}{4 \sqrt{3}}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 x\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 x\right )\\ &=-\frac{1}{x}+\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{1}{3} \tan ^{-1}(x)-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}+2 x\right )-\frac{\log \left (1-\sqrt{3} x+x^2\right )}{4 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} x+x^2\right )}{4 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0207184, size = 82, normalized size = 0.96 \[ -\frac{\sqrt{3} x \log \left (x^2-\sqrt{3} x+1\right )-\sqrt{3} x \log \left (x^2+\sqrt{3} x+1\right )-2 x \tan ^{-1}\left (\sqrt{3}-2 x\right )+4 x \tan ^{-1}(x)+2 x \tan ^{-1}\left (2 x+\sqrt{3}\right )+12}{12 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 66, normalized size = 0.8 \begin{align*} -{x}^{-1}-{\frac{\arctan \left ( x \right ) }{3}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50672, size = 88, normalized size = 1.04 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) - \frac{1}{x} - \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) - \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) - \frac{1}{3} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57757, size = 298, normalized size = 3.51 \begin{align*} \frac{\sqrt{3} x \log \left (x^{2} + \sqrt{3} x + 1\right ) - \sqrt{3} x \log \left (x^{2} - \sqrt{3} x + 1\right ) - 4 \, x \arctan \left (x\right ) + 4 \, x \arctan \left (-2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}\right ) + 4 \, x \arctan \left (-2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}\right ) - 12}{12 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.213678, size = 71, normalized size = 0.84 \begin{align*} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} - \frac{\operatorname{atan}{\left (x \right )}}{3} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{6} + 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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