Optimal. Leaf size=135 \[ -\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.12, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1423, 1161, 618, 204, 1164, 628} \[ -\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 1161
Rule 1164
Rule 1423
Rubi steps
\begin {align*} \int \frac {-1+\sqrt {3}+2 x^4}{1-x^4+x^8} \, dx &=\frac {\int \frac {\sqrt {3} \left (-1+\sqrt {3}\right )+\left (3-\sqrt {3}\right ) x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3} \left (-1+\sqrt {3}\right )+\left (-3+\sqrt {3}\right ) x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=-\frac {\int \frac {\sqrt {2-\sqrt {3}}+2 x}{-1-\sqrt {2-\sqrt {3}} x-x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2-\sqrt {3}}-2 x}{-1+\sqrt {2-\sqrt {3}} x-x^2} \, dx}{2 \sqrt {2}}+\frac {1}{4} \left (-1+\sqrt {3}\right ) \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{4} \left (-1+\sqrt {3}\right ) \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx\\ &=-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{2 \sqrt {2}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{2 \sqrt {2}}+\frac {1}{2} \left (1-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )+\frac {1}{2} \left (1-\sqrt {3}\right ) \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 )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{2 \sqrt {2}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 71, normalized size = 0.53 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {2 \text {$\#$1}^4 \log (x-\text {$\#$1})+\sqrt {3} \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 104, normalized size = 0.77 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} x^{3} - \sqrt {2} x\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {3} \sqrt {2} + \sqrt {2}\right )} x\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {{\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} x + 2 \, x^{2} + 2}{{\left (\sqrt {3} \sqrt {2} - \sqrt {2}\right )} x - 2 \, x^{2} - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 107, normalized size = 0.79 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 47, normalized size = 0.35 \[ \frac {\left (2 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{4}-1+\sqrt {3}\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{7}-4 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{3}} \]
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 \[ \int \frac {2 \, x^{4} + \sqrt {3} - 1}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.24, size = 133, normalized size = 0.99 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {72\,\sqrt {2}\,x}{144\,\sqrt {3}-144\,\sqrt {3}\,x^2-288\,x^2+288}+\frac {72\,\sqrt {2}\,\sqrt {3}\,x}{144\,\sqrt {3}-144\,\sqrt {3}\,x^2-288\,x^2+288}\right )}{2}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {72\,\sqrt {2}\,x}{144\,\sqrt {3}+144\,\sqrt {3}\,x^2+288\,x^2+288}+\frac {72\,\sqrt {2}\,\sqrt {3}\,x}{144\,\sqrt {3}+144\,\sqrt {3}\,x^2+288\,x^2+288}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 163, normalized size = 1.21 \[ \frac {\sqrt {2} \left (2 \operatorname {atan}{\left (x \left (\frac {\sqrt {6}}{1 + \sqrt {3}} + \frac {2 \sqrt {2}}{1 + \sqrt {3}}\right ) \right )} + 2 \operatorname {atan}{\left (x^{3} \left (\frac {\sqrt {6}}{1 + \sqrt {3}} + \frac {2 \sqrt {2}}{1 + \sqrt {3}}\right ) - \sqrt {2} x \right )}\right )}{4} - \frac {\sqrt {2} \log {\left (x^{2} - \frac {\sqrt {2} x \left (\frac {2}{\sqrt {3} + 2} + \frac {2 \sqrt {3}}{\sqrt {3} + 2}\right )}{4} + 1 \right )}}{4} + \frac {\sqrt {2} \log {\left (x^{2} + \frac {\sqrt {2} x \left (\frac {2}{\sqrt {3} + 2} + \frac {2 \sqrt {3}}{\sqrt {3} + 2}\right )}{4} + 1 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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