Optimal. Leaf size=90 \[ -\frac{1}{7 x^7}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{x}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
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Rubi [A] time = 0.247353, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{1}{7 x^7}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{x}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]
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^8 \left (1+x^6\right )} \, dx &=-\frac{1}{7 x^7}-\int \frac{1}{x^2 \left (1+x^6\right )} \, dx\\ &=-\frac{1}{7 x^7}+\frac{1}{x}+\int \frac{x^4}{1+x^6} \, dx\\ &=-\frac{1}{7 x^7}+\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}{7 x^7}+\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}{7 x^7}+\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}{7 x^7}+\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 = 0.0278923, size = 84, normalized size = 0.93 \[ \frac{1}{84} \left (-\frac{12}{x^7}+7 \sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )-7 \sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )+\frac{84}{x}-14 \tan ^{-1}\left (\sqrt{3}-2 x\right )+28 \tan ^{-1}(x)+14 \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 69, normalized size = 0.8 \begin{align*} -{\frac{1}{7\,{x}^{7}}}+{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 = 1.81703, size = 97, normalized size = 1.08 \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{7 \, x^{6} - 1}{7 \, x^{7}} + \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 = 1.56718, size = 338, normalized size = 3.76 \begin{align*} -\frac{7 \, \sqrt{3} x^{7} \log \left (x^{2} + \sqrt{3} x + 1\right ) - 7 \, \sqrt{3} x^{7} \log \left (x^{2} - \sqrt{3} x + 1\right ) - 28 \, x^{7} \arctan \left (x\right ) + 28 \, x^{7} \arctan \left (-2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}\right ) + 28 \, x^{7} \arctan \left (-2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}\right ) - 84 \, x^{6} + 12}{84 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.248719, size = 80, normalized size = 0.89 \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{7 x^{6} - 1}{7 x^{7}} \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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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