Optimal. Leaf size=173 \[ -\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]
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Rubi [A] time = 0.08, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1374, 212, 206, 203} \[ -\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 1374
Rubi steps
\begin {align*} \int \frac {x^4}{1-3 x^4+x^8} \, 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}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=\frac {1}{2} \sqrt {\frac {1}{5} \left (3-\sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx+\frac {1}{2} \sqrt {\frac {1}{5} \left (3-\sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx-\frac {1}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {5}\right )} \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx-\frac {1}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {5}\right )} \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx\\ &=-\frac {\sqrt [4]{\frac {1}{2} \left (3+\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 (3-\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 (3+\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 (3-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 132, normalized size = 0.76 \[ \frac {\sqrt {\sqrt {5}-1} \tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )-\sqrt {1+\sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {\sqrt {5}-1} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )-\sqrt {1+\sqrt {5}} \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 271, normalized size = 1.57 \[ -\frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \arctan \left (\frac {1}{40} \, \sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} + 1} {\left (\sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} - \frac {1}{20} \, \sqrt {10} {\left (\sqrt {5} x - 5 \, x\right )} \sqrt {\sqrt {5} + 1}\right ) - \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {1}{40} \, \sqrt {10} \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (\sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {5} - 1} - \frac {1}{20} \, \sqrt {10} {\left (\sqrt {5} x + 5 \, x\right )} \sqrt {\sqrt {5} - 1}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} + 1} + 10 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (-\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} + 1} + 10 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} - 1} + 10 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (-\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} - 1} + 10 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 147, normalized size = 0.85 \[ -\frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 206, normalized size = 1.19 \[ \frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}-\frac {\sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}-\frac {\arctanh \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{2 \sqrt {-2+2 \sqrt {5}}}-\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{2 \sqrt {2+2 \sqrt {5}}} \]
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 \[ \int \frac {x^{4}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 269, normalized size = 1.55 \[ \frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}-1\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}-1\right )}\right )\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}+1\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}+1\right )}\right )\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}-1\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}-1\right )}\right )\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}+1\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}+1\right )}\right )\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 49, normalized size = 0.28 \[ \operatorname {RootSum} {\left (6400 t^{4} - 80 t^{2} - 1, \left (t \mapsto t \log {\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 80 t^{2} - 1, \left (t \mapsto t \log {\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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