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 = 0.08, antiderivative size = 182, normalized size of antiderivative = 1.00, 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 = {1382, 1436,
218, 212, 209} \begin {gather*} -\frac {\sqrt [4]{\frac {1}{2} \left (843-377 \sqrt {5}\right )} \text {ArcTan}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\left (3+\sqrt {5}\right )^{7/4} \text {ArcTan}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {1}{3 x^3}-\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 1382
Rule 1436
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*}
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Mathematica [A]
time = 0.16, size = 166, normalized size = 0.91 \begin {gather*} -\frac {1}{3 x^3}+\frac {\left (2+\sqrt {5}\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 ) \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}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (-2+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 135, normalized size = 0.74
method | result | size |
risch | \(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}-145 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (35 \textit {\_R}^{3}-199 \textit {\_R} +13 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+145 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-35 \textit {\_R}^{3}-199 \textit {\_R} +13 x \right )\right )}{4}\) | \(73\) |
default | \(\frac {\sqrt {5}\, \left (2+\sqrt {5}\right ) \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}-\frac {\left (-2+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}-\frac {\left (-2+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \left (2+\sqrt {5}\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}-\frac {1}{3 x^{3}}\) | \(135\) |
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. 335 vs.
\(2 (128) = 256\).
time = 0.37, size = 335, normalized size = 1.84 \begin {gather*} \frac {12 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} - 29} \arctan \left (\frac {1}{20} \, \sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} + 1} \sqrt {13 \, \sqrt {5} - 29} {\left (2 \, \sqrt {5} + 5\right )} - \frac {1}{10} \, \sqrt {10} {\left (2 \, \sqrt {5} x + 5 \, x\right )} \sqrt {13 \, \sqrt {5} - 29}\right ) + 12 \, \sqrt {10} x^{3} \sqrt {13 \, \sqrt {5} + 29} \arctan \left (\frac {1}{20} \, \sqrt {10} \sqrt {2} \sqrt {2 \, x^{2} + \sqrt {5} - 1} \sqrt {13 \, \sqrt {5} + 29} {\left (2 \, \sqrt {5} - 5\right )} - \frac {1}{10} \, \sqrt {10} {\left (2 \, \sqrt {5} x - 5 \, x\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.77, size = 63, normalized size = 0.35 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.42, size = 152, normalized size = 0.84 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 268, normalized size = 1.47 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x\,\sqrt {-130\,\sqrt {5}-290}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}+196417\right )}+\frac {\sqrt {5}\,x\,\sqrt {-130\,\sqrt {5}-290}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}+196417\right )}\right )\,\sqrt {-130\,\sqrt {5}-290}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {290-130\,\sqrt {5}}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}-196417\right )}-\frac {\sqrt {5}\,x\,\sqrt {290-130\,\sqrt {5}}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}-196417\right )}\right )\,\sqrt {290-130\,\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {1}{3\,x^3}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}-29}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}-196417\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}-29}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}-196417\right )}\right )\,\sqrt {13\,\sqrt {5}-29}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}+29}\,20735{}\mathrm {i}}{2\,\left (87841\,\sqrt {5}+196417\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {13\,\sqrt {5}+29}\,46371{}\mathrm {i}}{10\,\left (87841\,\sqrt {5}+196417\right )}\right )\,\sqrt {13\,\sqrt {5}+29}\,1{}\mathrm {i}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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