Optimal. Leaf size=355 \[ \frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1373, 1094, 634, 618, 204, 628} \[ \frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1094
Rule 1373
Rubi steps
\begin {align*} \int \frac {x^2}{1-x^4+x^8} \, dx &=\frac {\int \frac {1}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {1}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=-\frac {\int \frac {\sqrt {2-\sqrt {3}}-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}}+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}}-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}}+x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ &=-\frac {\int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}-\frac {\int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \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}{8 \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}{8 \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}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ &=\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \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 )}{4 \sqrt {3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )}{4 \sqrt {3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )}{4 \sqrt {3}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.11 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^5-\text {$\#$1}}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 567, normalized size = 1.60 \[ -\frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (2 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 12 \, x^{2} + 12\right ) + \frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (-2 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 12 \, x^{2} + 12\right ) - \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (\sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 12 \, x^{2} + 12\right ) + \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (-\sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 12 \, x^{2} + 12\right ) - \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + \frac {1}{6} \, \sqrt {6} \sqrt {2} \sqrt {2 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 12 \, x^{2} + 12} \sqrt {\sqrt {3} + 2} - \sqrt {3} - 2\right ) - \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {3} + 2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + \frac {1}{6} \, \sqrt {6} \sqrt {2} \sqrt {-2 \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {\sqrt {3} + 2} + 12 \, x^{2} + 12} \sqrt {\sqrt {3} + 2} + \sqrt {3} + 2\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {2} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-\frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 12 \, x^{2} + 12} \sqrt {-4 \, \sqrt {3} + 8} + \sqrt {3} - 2\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {2} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-\frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {-\sqrt {6} \sqrt {3} \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 8} + 12 \, x^{2} + 12} \sqrt {-4 \, \sqrt {3} + 8} - \sqrt {3} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 253, normalized size = 0.71 \[ \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 40, normalized size = 0.11 \[ \frac {\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{2} \ln \left (-\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{7}-4 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 286, normalized size = 0.81 \[ -\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {2^{3/4}\,\sqrt {3}\,x\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.28, size = 26, normalized size = 0.07 \[ \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left (t \mapsto t \log {\left (- 442368 t^{7} - 192 t^{3} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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