Optimal. Leaf size=182 \[ -\frac{1}{3 x^3}-\frac{\sqrt [4]{\frac{1}{2} \left (843-377 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{7/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (843-377 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{7/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4\ 2^{3/4} \sqrt{5}} \]
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Rubi [A] time = 0.117854, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1368, 1422, 212, 206, 203} \[ -\frac{1}{3 x^3}-\frac{\sqrt [4]{\frac{1}{2} \left (843-377 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{7/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (843-377 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{7/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4\ 2^{3/4} \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1368
Rule 1422
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (1-3 x^4+x^8\right )} \, dx &=-\frac{1}{3 x^3}+\frac{1}{3} \int \frac{9-3 x^4}{1-3 x^4+x^8} \, dx\\ &=-\frac{1}{3 x^3}+\frac{1}{10} \left (-5+3 \sqrt{5}\right ) \int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx-\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{1}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx\\ &=-\frac{1}{3 x^3}+\frac{\left (5-3 \sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2} \, dx}{10 \sqrt{3+\sqrt{5}}}+\frac{\left (5-3 \sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2} \, dx}{10 \sqrt{3+\sqrt{5}}}+\frac{\left (3+\sqrt{5}\right )^{3/2} \int \frac{1}{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2} \, dx}{4 \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{3/2} \int \frac{1}{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2} \, dx}{4 \sqrt{5}}\\ &=-\frac{1}{3 x^3}-\frac{\sqrt [4]{\frac{1}{2} \left (843-377 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (843+377 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (843-377 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (843+377 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.267259, size = 166, normalized size = 0.91 \[ -\frac{1}{3 x^3}+\frac{\left (2+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (\sqrt{5}-2\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\left (2+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (\sqrt{5}-2\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 209, normalized size = 1.2 \begin{align*}{\frac{2\,\sqrt{5}}{5\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{2\,\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{2\,\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{2\,\sqrt{5}}{5\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{3\,{x}^{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*} -\frac{1}{3 \, x^{3}} - \frac{1}{2} \, \int \frac{2 \, x^{2} + 3}{x^{4} + x^{2} - 1}\,{d x} + \frac{1}{2} \, \int \frac{2 \, x^{2} - 3}{x^{4} - x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90534, size = 1037, normalized size = 5.7 \begin{align*} \frac{12 \, \sqrt{10} x^{3} \sqrt{13 \, \sqrt{5} + 29} \arctan \left (\frac{1}{20} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (2 \, \sqrt{5} \sqrt{2} - 5 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (2 \, \sqrt{5} x - 5 \, x\right )}\right )} \sqrt{13 \, \sqrt{5} + 29}\right ) + 12 \, \sqrt{10} x^{3} \sqrt{13 \, \sqrt{5} - 29} \arctan \left (\frac{1}{20} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (2 \, \sqrt{5} \sqrt{2} + 5 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (2 \, \sqrt{5} x + 5 \, x\right )}\right )} \sqrt{13 \, \sqrt{5} - 29}\right ) - 3 \, \sqrt{10} x^{3} \sqrt{13 \, \sqrt{5} - 29} \log \left (\sqrt{10} \sqrt{13 \, \sqrt{5} - 29}{\left (7 \, \sqrt{5} + 15\right )} + 20 \, x\right ) + 3 \, \sqrt{10} x^{3} \sqrt{13 \, \sqrt{5} - 29} \log \left (-\sqrt{10} \sqrt{13 \, \sqrt{5} - 29}{\left (7 \, \sqrt{5} + 15\right )} + 20 \, x\right ) + 3 \, \sqrt{10} x^{3} \sqrt{13 \, \sqrt{5} + 29} \log \left (\sqrt{10} \sqrt{13 \, \sqrt{5} + 29}{\left (7 \, \sqrt{5} - 15\right )} + 20 \, x\right ) - 3 \, \sqrt{10} x^{3} \sqrt{13 \, \sqrt{5} + 29} \log \left (-\sqrt{10} \sqrt{13 \, \sqrt{5} + 29}{\left (7 \, \sqrt{5} - 15\right )} + 20 \, x\right ) - 40}{120 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.974728, size = 63, normalized size = 0.35 \begin{align*} \operatorname{RootSum}{\left (6400 t^{4} - 2320 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{179200 t^{5}}{377} - \frac{23112 t}{377} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 2320 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{179200 t^{5}}{377} - \frac{23112 t}{377} + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22524, size = 205, normalized size = 1.13 \begin{align*} -\frac{1}{20} \, \sqrt{130 \, \sqrt{5} - 290} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{20} \, \sqrt{130 \, \sqrt{5} + 290} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{130 \, \sqrt{5} - 290} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{130 \, \sqrt{5} - 290} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{130 \, \sqrt{5} + 290} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{130 \, \sqrt{5} + 290} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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