Optimal. Leaf size=331 \[ -\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {2-\sqrt {3}}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {2-\sqrt {3}}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {2+\sqrt {3}}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {2+\sqrt {3}}}-\frac {1}{4} \sqrt {2-\sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {3}} \tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1419, 1094, 634, 618, 204, 628} \[ -\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {2-\sqrt {3}}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{8 \sqrt {2-\sqrt {3}}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {2+\sqrt {3}}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{8 \sqrt {2+\sqrt {3}}}-\frac {1}{4} \sqrt {2-\sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {3}} \tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1094
Rule 1419
Rubi steps
\begin {align*} \int \frac {1+x^4}{1-x^4+x^8} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {3} x^2+x^4} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {3} x^2+x^4} \, dx\\ &=\frac {\int \frac {\sqrt {2-\sqrt {3}}-x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\sqrt {2-\sqrt {3}}+x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}-x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}+x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {2+\sqrt {3}}}\\ &=\frac {1}{8} \int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx-\frac {\int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {2-\sqrt {3}}}-\frac {\int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{8 \sqrt {2+\sqrt {3}}}\\ &=-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {2-\sqrt {3}}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {2-\sqrt {3}}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {2+\sqrt {3}}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {2+\sqrt {3}}}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {2+\sqrt {3}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {2-\sqrt {3}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {2+\sqrt {3}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {2-\sqrt {3}}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {2-\sqrt {3}}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {2-\sqrt {3}}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {2+\sqrt {3}}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {2+\sqrt {3}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 55, normalized size = 0.17 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+\log (x-\text {$\#$1})}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 377, normalized size = 1.14 \[ -\frac {1}{8} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (2 \, x^{2} + 2 \, x \sqrt {\sqrt {3} + 2} + 2\right ) + \frac {1}{8} \, \sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} \log \left (2 \, x^{2} - 2 \, x \sqrt {\sqrt {3} + 2} + 2\right ) + \frac {1}{16} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (2 \, x^{2} + x \sqrt {-4 \, \sqrt {3} + 8} + 2\right ) - \frac {1}{16} \, {\left (\sqrt {3} + 2\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (2 \, x^{2} - x \sqrt {-4 \, \sqrt {3} + 8} + 2\right ) - \frac {1}{2} \, \sqrt {\sqrt {3} + 2} \arctan \left (\sqrt {2} \sqrt {2 \, x^{2} + 2 \, x \sqrt {\sqrt {3} + 2} + 2} \sqrt {\sqrt {3} + 2} - 2 \, x \sqrt {\sqrt {3} + 2} - \sqrt {3} - 2\right ) - \frac {1}{2} \, \sqrt {\sqrt {3} + 2} \arctan \left (\sqrt {2} \sqrt {2 \, x^{2} - 2 \, x \sqrt {\sqrt {3} + 2} + 2} \sqrt {\sqrt {3} + 2} - 2 \, x \sqrt {\sqrt {3} + 2} + \sqrt {3} + 2\right ) - \frac {1}{4} \, \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x^{2} + x \sqrt {-4 \, \sqrt {3} + 8} + 2} \sqrt {-4 \, \sqrt {3} + 8} - x \sqrt {-4 \, \sqrt {3} + 8} + \sqrt {3} - 2\right ) - \frac {1}{4} \, \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x^{2} - x \sqrt {-4 \, \sqrt {3} + 8} + 2} \sqrt {-4 \, \sqrt {3} + 8} - x \sqrt {-4 \, \sqrt {3} + 8} - \sqrt {3} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 245, normalized size = 0.74 \[ \frac {1}{8} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{16} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{16} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{16} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{16} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 42, normalized size = 0.13 \[ \frac {\left (\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{7}-4 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{3}} \]
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 \[ \int \frac {x^{4} + 1}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 145, normalized size = 0.44 \[ -\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27-27{}\mathrm {i}\right )}{27\,\sqrt {3}-81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )\right )-\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27+27{}\mathrm {i}\right )}{27\,\sqrt {3}-81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )\right )-\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27-27{}\mathrm {i}\right )}{27\,\sqrt {3}+81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )\right )-\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27+27{}\mathrm {i}\right )}{27\,\sqrt {3}+81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.10, size = 20, normalized size = 0.06 \[ \operatorname {RootSum} {\left (65536 t^{8} - 256 t^{4} + 1, \left (t \mapsto t \log {\left (1024 t^{5} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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