Optimal. Leaf size=171 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}} \]
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Rubi [A] time = 0.151257, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1419, 1093, 203, 207} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{6 \left (\sqrt{7}-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}} \]
Antiderivative was successfully verified.
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Rule 1419
Rule 1093
Rule 203
Rule 207
Rubi steps
\begin{align*} \int \frac{1+x^4}{1-5 x^4+x^8} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{7} x^2+x^4} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{7} x^2+x^4} \, dx\\ &=\frac{\int \frac{1}{-\frac{\sqrt{3}}{2}-\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}-\frac{\int \frac{1}{\frac{\sqrt{3}}{2}-\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}+\frac{\int \frac{1}{-\frac{\sqrt{3}}{2}+\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}-\frac{\int \frac{1}{\frac{\sqrt{3}}{2}+\frac{\sqrt{7}}{2}+x^2} \, dx}{2 \sqrt{3}}\\ &=\frac{\tan ^{-1}\left (\sqrt{\frac{2}{-\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (-\sqrt{3}+\sqrt{7}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{-\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (-\sqrt{3}+\sqrt{7}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{6 \left (\sqrt{3}+\sqrt{7}\right )}}\\ \end{align*}
Mathematica [C] time = 0.0132405, size = 55, normalized size = 0.32 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-5 \text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-5 \text{$\#$1}^3}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 42, 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 = 1.45293, size = 1812, normalized size = 10.6 \begin{align*} \frac{1}{6} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \arctan \left (\frac{1}{48} \,{\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} + 3 \, \sqrt{6} \sqrt{2}\right )} \sqrt{4 \, x^{2} +{\left (\sqrt{7} \sqrt{3} \sqrt{2} + 5 \, \sqrt{2}\right )} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \sqrt{-\sqrt{7} \sqrt{3} + 5} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} - \frac{1}{24} \,{\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} x + 3 \, \sqrt{6} \sqrt{2} x\right )} \sqrt{-\sqrt{7} \sqrt{3} + 5} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}}\right ) - \frac{1}{6} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \arctan \left (\frac{1}{48} \,{\left ({\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} - 3 \, \sqrt{6} \sqrt{2}\right )} \sqrt{4 \, x^{2} -{\left (\sqrt{7} \sqrt{3} \sqrt{2} - 5 \, \sqrt{2}\right )} \sqrt{\sqrt{7} \sqrt{3} + 5}} \sqrt{\sqrt{7} \sqrt{3} + 5} - 2 \,{\left (\sqrt{7} \sqrt{6} \sqrt{3} \sqrt{2} x - 3 \, \sqrt{6} \sqrt{2} x\right )} \sqrt{\sqrt{7} \sqrt{3} + 5}\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}}\right ) + \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \log \left ({\left (\sqrt{7} \sqrt{6} \sqrt{3} - 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) - \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} \log \left (-{\left (\sqrt{7} \sqrt{6} \sqrt{3} - 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) - \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \log \left ({\left (\sqrt{7} \sqrt{6} \sqrt{3} + 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) + \frac{1}{24} \, \sqrt{6} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} \log \left (-{\left (\sqrt{7} \sqrt{6} \sqrt{3} + 3 \, \sqrt{6}\right )} \sqrt{\sqrt{2} \sqrt{-\sqrt{7} \sqrt{3} + 5}} + 12 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.190294, size = 24, normalized size = 0.14 \begin{align*} \operatorname{RootSum}{\left (5308416 t^{8} - 11520 t^{4} + 1, \left ( t \mapsto t \log{\left (9216 t^{5} - 16 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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