Optimal. Leaf size=169 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}} \]
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Rubi [A] time = 0.142447, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1419, 1093, 207, 203} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}} \]
Antiderivative was successfully verified.
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Rule 1419
Rule 1093
Rule 207
Rule 203
Rubi steps
\begin{align*} \int \frac{1-x^4}{1-5 x^4+x^8} \, dx &=-\left (\frac{1}{2} \int \frac{1}{-1-\sqrt{3} x^2+x^4} \, dx\right )-\frac{1}{2} \int \frac{1}{-1+\sqrt{3} x^2+x^4} \, dx\\ &=-\frac{\int \frac{1}{-\frac{\sqrt{3}}{2}-\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{7}}-\frac{\int \frac{1}{\frac{\sqrt{3}}{2}-\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{7}}+\frac{\int \frac{1}{-\frac{\sqrt{3}}{2}+\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{7}}+\frac{\int \frac{1}{\frac{\sqrt{3}}{2}+\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{7}}\\ &=\frac{\tan ^{-1}\left (\sqrt{\frac{2}{-\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (-\sqrt{3}+\sqrt{7}\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{-\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (-\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}\\ \end{align*}
Mathematica [C] time = 0.0133492, size = 57, normalized size = 0.34 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-5 \text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-5 \text{$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 44, normalized size = 0.3 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-5\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-5\,{{\it \_R}}^{3}}}} \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{x^{4} - 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55648, size = 1751, normalized size = 10.36 \begin{align*} -\frac{1}{14} \, \sqrt{14} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \arctan \left (\frac{1}{112} \, \sqrt{14} \sqrt{4 \, x^{2} +{\left (\sqrt{7} \sqrt{3} \sqrt{2} + 5 \, \sqrt{2}\right )} \sqrt{-\sqrt{7} \sqrt{3} + 5}}{\left (\sqrt{7} \sqrt{3} \sqrt{2} + 7 \, \sqrt{2}\right )} \sqrt{-\sqrt{7} \sqrt{3} + 5} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} - \frac{1}{56} \, \sqrt{14}{\left (\sqrt{7} \sqrt{3} \sqrt{2} x + 7 \, \sqrt{2} x\right )} \sqrt{-\sqrt{7} \sqrt{3} + 5} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}}\right ) + \frac{1}{14} \, \sqrt{14} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \arctan \left (\frac{1}{112} \,{\left (\sqrt{14} \sqrt{4 \, x^{2} -{\left (\sqrt{7} \sqrt{3} \sqrt{2} - 5 \, \sqrt{2}\right )} \sqrt{\sqrt{7} \sqrt{3} + 5}}{\left (\sqrt{7} \sqrt{3} \sqrt{2} - 7 \, \sqrt{2}\right )} \sqrt{\sqrt{7} \sqrt{3} + 5} - 2 \, \sqrt{14}{\left (\sqrt{7} \sqrt{3} \sqrt{2} x - 7 \, \sqrt{2} x\right )} \sqrt{\sqrt{7} \sqrt{3} + 5}\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}}\right ) - \frac{1}{56} \, \sqrt{14} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \log \left (\sqrt{14}{\left (\sqrt{7} \sqrt{3} - 7\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} + 28 \, x\right ) + \frac{1}{56} \, \sqrt{14} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \log \left (-\sqrt{14}{\left (\sqrt{7} \sqrt{3} - 7\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} + 28 \, x\right ) + \frac{1}{56} \, \sqrt{14} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \log \left (\sqrt{14}{\left (\sqrt{7} \sqrt{3} + 7\right )} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} + 28 \, x\right ) - \frac{1}{56} \, \sqrt{14} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \log \left (-\sqrt{14}{\left (\sqrt{7} \sqrt{3} + 7\right )} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} + 28 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.185546, size = 26, normalized size = 0.15 \begin{align*} - \operatorname{RootSum}{\left (157351936 t^{8} - 62720 t^{4} + 1, \left ( t \mapsto t \log{\left (50176 t^{5} - 24 t + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{4} - 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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