Optimal. Leaf size=275 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}} \]
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Rubi [A] time = 0.262004, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1346, 1169, 634, 618, 204, 628} \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}} \]
Antiderivative was successfully verified.
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Rule 1346
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{1-x^4+x^8} \, dx &=\frac{\int \frac{\sqrt{3}-x^2}{1-\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}+\frac{\int \frac{\sqrt{3}+x^2}{1+\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}\\ &=\frac{\int \frac{\sqrt{3 \left (2-\sqrt{3}\right )}-\left (-1+\sqrt{3}\right ) x}{1-\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\int \frac{\sqrt{3 \left (2-\sqrt{3}\right )}+\left (-1+\sqrt{3}\right ) x}{1+\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\int \frac{\sqrt{3 \left (2+\sqrt{3}\right )}-\left (1+\sqrt{3}\right ) x}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\int \frac{\sqrt{3 \left (2+\sqrt{3}\right )}+\left (1+\sqrt{3}\right ) x}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}\\ &=-\frac{\int \frac{-\sqrt{2-\sqrt{3}}+2 x}{1-\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6}}+\frac{\int \frac{\sqrt{2-\sqrt{3}}+2 x}{1+\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6}}-\frac{\int \frac{-\sqrt{2+\sqrt{3}}+2 x}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6}}+\frac{\int \frac{\sqrt{2+\sqrt{3}}+2 x}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6}}+\frac{\int \frac{1}{1-\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6 \left (2-\sqrt{3}\right )}}+\frac{\int \frac{1}{1+\sqrt{2-\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6 \left (2-\sqrt{3}\right )}}+\frac{\int \frac{1}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6 \left (2+\sqrt{3}\right )}}+\frac{\int \frac{1}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{6 \left (2+\sqrt{3}\right )}}\\ &=-\frac{\log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}+\frac{\log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}-\frac{\log \left (1-\sqrt{2+\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}+\frac{\log \left (1+\sqrt{2+\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{3}-x^2} \, dx,x,-\sqrt{2-\sqrt{3}}+2 x\right )}{2 \sqrt{6 \left (2-\sqrt{3}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{3}-x^2} \, dx,x,\sqrt{2-\sqrt{3}}+2 x\right )}{2 \sqrt{6 \left (2-\sqrt{3}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,-\sqrt{2+\sqrt{3}}+2 x\right )}{2 \sqrt{6 \left (2+\sqrt{3}\right )}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,\sqrt{2+\sqrt{3}}+2 x\right )}{2 \sqrt{6 \left (2+\sqrt{3}\right )}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}+2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}+2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}+\frac{\log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}-\frac{\log \left (1-\sqrt{2+\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}+\frac{\log \left (1+\sqrt{2+\sqrt{3}} x+x^2\right )}{4 \sqrt{6}}\\ \end{align*}
Mathematica [C] time = 0.0117178, size = 42, normalized size = 0.15 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 30, normalized size = 0.1 \begin{align*}{\frac{\sum _{{\it \_R}={\it RootOf} \left ( 9\,{{\it \_Z}}^{4}+1 \right ) }{\it \_R}\,\ln \left ( 3\,{{\it \_R}}^{2}+3\,{\it \_R}\,x+{x}^{2} \right ) }{4}} \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*} \int \frac{1}{x^{8} - x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13769, size = 608, normalized size = 2.21 \begin{align*} -\frac{1}{6} \, \sqrt{3} \sqrt{2} \arctan \left (-\frac{\sqrt{3} \sqrt{2}{\left (x^{3} - x\right )} + x^{2} - \sqrt{x^{4} + \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1}{\left (\sqrt{3} \sqrt{2} x - 2\right )}}{3 \, x^{2} - 2}\right ) - \frac{1}{6} \, \sqrt{3} \sqrt{2} \arctan \left (-\frac{\sqrt{3} \sqrt{2}{\left (x^{3} - x\right )} - x^{2} - \sqrt{x^{4} - \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1}{\left (\sqrt{3} \sqrt{2} x + 2\right )}}{3 \, x^{2} - 2}\right ) + \frac{1}{24} \, \sqrt{3} \sqrt{2} \log \left (x^{4} + \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) - \frac{1}{24} \, \sqrt{3} \sqrt{2} \log \left (x^{4} - \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.203264, size = 165, normalized size = 0.6 \begin{align*} \frac{\sqrt{6} \left (2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{3} - \frac{1}{3} \right )} + 2 \operatorname{atan}{\left (\sqrt{6} x^{3} - 4 x^{2} + 2 \sqrt{6} x - 3 \right )}\right )}{24} + \frac{\sqrt{6} \left (2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{3} + \frac{1}{3} \right )} + 2 \operatorname{atan}{\left (\sqrt{6} x^{3} + 4 x^{2} + 2 \sqrt{6} x + 3 \right )}\right )}{24} - \frac{\sqrt{6} \log{\left (x^{4} - \sqrt{6} x^{3} + 3 x^{2} - \sqrt{6} x + 1 \right )}}{24} + \frac{\sqrt{6} \log{\left (x^{4} + \sqrt{6} x^{3} + 3 x^{2} + \sqrt{6} x + 1 \right )}}{24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09782, size = 277, normalized size = 1.01 \begin{align*} \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{24} \, \sqrt{6} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{24} \, \sqrt{6} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{24} \, \sqrt{6} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{24} \, \sqrt{6} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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