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 = 0.05, 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 = {1388, 218, 212,
209} \begin {gather*} -\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {ArcTan}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {ArcTan}\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 {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 1388
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 = 0.12, size = 132, normalized size = 0.76 \begin {gather*} \frac {\sqrt {-1+\sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {1+\sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {-1+\sqrt {5}} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {1+\sqrt {5}} \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 130, normalized size = 0.75
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-10 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (10 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}\) | \(60\) |
default | \(\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}\) | \(130\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 261 vs.
\(2 (119) = 238\).
time = 0.38, size = 261, normalized size = 1.51 \begin {gather*} -\frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {1}{40} \, \sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} - 1} {\left (\sqrt {5} + 5\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} \sqrt {2 \, x^{2} + \sqrt {5} + 1} \sqrt {\sqrt {5} + 1} {\left (\sqrt {5} - 5\right )} - \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.71, size = 49, normalized size = 0.28 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.88, size = 147, normalized size = 0.85 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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