Integrand size = 16, antiderivative size = 189 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]
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Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1382, 1518, 1436, 218, 212, 209} \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (39603+17711 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {1}{7 x^7}-\frac {1}{x^3} \]
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Rule 209
Rule 212
Rule 218
Rule 1382
Rule 1436
Rule 1518
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{7 x^7}+\frac {1}{7} \int \frac {21-7 x^4}{x^4 \left (1-3 x^4+x^8\right )} \, dx \\ & = -\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {1}{21} \int \frac {-168+63 x^4}{1-3 x^4+x^8} \, dx \\ & = -\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {1}{10} \left (15-7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx-\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx \\ & = -\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {\left (-15+7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}-\frac {\left (-15+7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}+\frac {1}{2} \sqrt {\frac {1}{5} \left (123+55 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx+\frac {1}{2} \sqrt {\frac {1}{5} \left (123+55 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx \\ & = -\frac {1}{7 x^7}-\frac {1}{x^3}-\frac {\sqrt [4]{\frac {1}{2} \left (39603-17711 \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 (39603+17711 \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 (39603-17711 \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 (39603+17711 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{7 x^7}-\frac {1}{x^3}+\frac {\left (11+5 \sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\left (11-5 \sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\left (-11-5 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (-11+5 \sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.42
method | result | size |
risch | \(\frac {-x^{4}-\frac {1}{7}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-995 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (90 \textit {\_R}^{3}-3571 \textit {\_R} +89 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+995 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-90 \textit {\_R}^{3}-3571 \textit {\_R} +89 x \right )\right )}{4}\) | \(79\) |
default | \(-\frac {1}{7 x^{7}}-\frac {1}{x^{3}}-\frac {\left (-11+5 \sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \left (11+5 \sqrt {5}\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\left (-11+5 \sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \left (11+5 \sqrt {5}\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}\) | \(148\) |
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Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (133) = 266\).
Time = 0.28 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.86 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} - 199} \log \left (\sqrt {10} \sqrt {89 \, \sqrt {5} - 199} {\left (9 \, \sqrt {5} + 20\right )} + 10 \, x\right ) - 7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} - 199} \log \left (-\sqrt {10} \sqrt {89 \, \sqrt {5} - 199} {\left (9 \, \sqrt {5} + 20\right )} + 10 \, x\right ) - 7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} + 199} \log \left (\sqrt {10} \sqrt {89 \, \sqrt {5} + 199} {\left (9 \, \sqrt {5} - 20\right )} + 10 \, x\right ) + 7 \, \sqrt {10} x^{7} \sqrt {89 \, \sqrt {5} + 199} \log \left (-\sqrt {10} \sqrt {89 \, \sqrt {5} + 199} {\left (9 \, \sqrt {5} - 20\right )} + 10 \, x\right ) + 7 \, \sqrt {10} x^{7} \sqrt {-89 \, \sqrt {5} + 199} \log \left (\sqrt {10} {\left (9 \, \sqrt {5} + 20\right )} \sqrt {-89 \, \sqrt {5} + 199} + 10 \, x\right ) - 7 \, \sqrt {10} x^{7} \sqrt {-89 \, \sqrt {5} + 199} \log \left (-\sqrt {10} {\left (9 \, \sqrt {5} + 20\right )} \sqrt {-89 \, \sqrt {5} + 199} + 10 \, x\right ) - 7 \, \sqrt {10} x^{7} \sqrt {-89 \, \sqrt {5} - 199} \log \left (\sqrt {10} {\left (9 \, \sqrt {5} - 20\right )} \sqrt {-89 \, \sqrt {5} - 199} + 10 \, x\right ) + 7 \, \sqrt {10} x^{7} \sqrt {-89 \, \sqrt {5} - 199} \log \left (-\sqrt {10} {\left (9 \, \sqrt {5} - 20\right )} \sqrt {-89 \, \sqrt {5} - 199} + 10 \, x\right ) + 280 \, x^{4} + 40}{280 \, x^{7}} \]
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Time = 0.80 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 15920 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {460800 t^{5}}{17711} - \frac {2842588 t}{17711} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 15920 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {460800 t^{5}}{17711} - \frac {2842588 t}{17711} + x \right )} \right )\right )} + \frac {- 7 x^{4} - 1}{7 x^{7}} \]
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\[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 3 \, x^{4} + 1\right )} x^{8}} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{20} \, \sqrt {890 \, \sqrt {5} - 1990} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {890 \, \sqrt {5} + 1990} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {890 \, \sqrt {5} - 1990} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {890 \, \sqrt {5} - 1990} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {890 \, \sqrt {5} + 1990} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {890 \, \sqrt {5} + 1990} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {7 \, x^{4} + 1}{7 \, x^{7}} \]
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Time = 8.38 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^8 \left (1-3 x^4+x^8\right )} \, dx=-\frac {x^4+\frac {1}{7}}{x^7}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}+165580139\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}+165580139\right )}\right )\,\sqrt {-89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {199-89\,\sqrt {5}}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {199-89\,\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}-165580139\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}-199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}-165580139\right )}\right )\,\sqrt {89\,\sqrt {5}-199}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,6677047{}\mathrm {i}}{2\,\left (74049691\,\sqrt {5}+165580139\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {89\,\sqrt {5}+199}\,14930373{}\mathrm {i}}{10\,\left (74049691\,\sqrt {5}+165580139\right )}\right )\,\sqrt {89\,\sqrt {5}+199}\,1{}\mathrm {i}}{20} \]
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