Optimal. Leaf size=96 \[ -\frac{1}{2 x^2}-\frac{1}{6 x^6}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )-\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right ) \]
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Rubi [A] time = 0.0943797, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {1359, 1123, 1281, 12, 1127, 1161, 618, 204, 1164, 628} \[ -\frac{1}{2 x^2}-\frac{1}{6 x^6}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )-\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right ) \]
Antiderivative was successfully verified.
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Rule 1359
Rule 1123
Rule 1281
Rule 12
Rule 1127
Rule 1161
Rule 618
Rule 204
Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (1-x^4+x^8\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1-x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{3-3 x^2}{x^2 \left (1-x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{1}{2 x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{3 x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{1}{2 x^2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{1}{2 x^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1+x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{1}{2 x^2}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,x^2\right )-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,x^2\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{-1-\sqrt{3} x-x^2} \, dx,x,x^2\right )}{8 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{3}-2 x}{-1+\sqrt{3} x-x^2} \, dx,x,x^2\right )}{8 \sqrt{3}}\\ &=-\frac{1}{6 x^6}-\frac{1}{2 x^2}-\frac{\log \left (1-\sqrt{3} x^2+x^4\right )}{8 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} x^2+x^4\right )}{8 \sqrt{3}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+2 x^2\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+2 x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{1}{2 x^2}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}+2 x^2\right )-\frac{\log \left (1-\sqrt{3} x^2+x^4\right )}{8 \sqrt{3}}+\frac{\log \left (1+\sqrt{3} x^2+x^4\right )}{8 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0175828, size = 56, normalized size = 0.58 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\& \right ]-\frac{1}{2 x^2}-\frac{1}{6 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 75, normalized size = 0.8 \begin{align*} -{\frac{1}{6\,{x}^{6}}}-{\frac{1}{2\,{x}^{2}}}-{\frac{\arctan \left ( 2\,{x}^{2}-\sqrt{3} \right ) }{4}}-{\frac{\arctan \left ( 2\,{x}^{2}+\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}} \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{3 \, x^{4} + 1}{6 \, x^{6}} - \int \frac{x^{5}}{x^{8} - x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57348, size = 576, normalized size = 6. \begin{align*} \frac{4 \, \sqrt{6} \sqrt{3} \sqrt{2} x^{6} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x^{2} + \frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2 \, x^{4} + \sqrt{6} \sqrt{2} x^{2} + 2} - \sqrt{3}\right ) + 4 \, \sqrt{6} \sqrt{3} \sqrt{2} x^{6} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x^{2} + \frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2 \, x^{4} - \sqrt{6} \sqrt{2} x^{2} + 2} + \sqrt{3}\right ) + \sqrt{6} \sqrt{2} x^{6} \log \left (2 \, x^{4} + \sqrt{6} \sqrt{2} x^{2} + 2\right ) - \sqrt{6} \sqrt{2} x^{6} \log \left (2 \, x^{4} - \sqrt{6} \sqrt{2} x^{2} + 2\right ) - 24 \, x^{4} - 8}{48 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.256651, size = 82, normalized size = 0.85 \begin{align*} - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} - \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} - \frac{3 x^{4} + 1}{6 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27311, size = 358, normalized size = 3.73 \begin{align*} -\frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) + \frac{1}{96} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{3 \, x^{4} + 1}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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