Integrand size = 13, antiderivative size = 93 \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=-\frac {\sqrt [4]{-1+x^4}}{4 x^4}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{-1+\sqrt {-1+x^4}}\right )}{8 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{8 \sqrt {2}} \]
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Time = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.56, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {272, 43, 65, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{8 \sqrt {2}}+\frac {\arctan \left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{8 \sqrt {2}}-\frac {\sqrt [4]{x^4-1}}{4 x^4}-\frac {\log \left (\sqrt {x^4-1}-\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{16 \sqrt {2}}+\frac {\log \left (\sqrt {x^4-1}+\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{16 \sqrt {2}} \]
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Rule 43
Rule 65
Rule 210
Rule 217
Rule 272
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [4]{-1+x}}{x^2} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{-1+x^4}}{4 x^4}+\frac {1}{16} \text {Subst}\left (\int \frac {1}{(-1+x)^{3/4} x} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{-1+x^4}}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right ) \\ & = -\frac {\sqrt [4]{-1+x^4}}{4 x^4}+\frac {1}{8} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{8} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right ) \\ & = -\frac {\sqrt [4]{-1+x^4}}{4 x^4}+\frac {1}{16} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{16} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{16 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{16 \sqrt {2}} \\ & = -\frac {\sqrt [4]{-1+x^4}}{4 x^4}-\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{16 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{16 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{8 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{8 \sqrt {2}} \\ & = -\frac {\sqrt [4]{-1+x^4}}{4 x^4}-\frac {\arctan \left (1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{8 \sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{8 \sqrt {2}}-\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{16 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{16 \sqrt {2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=\frac {1}{16} \left (-\frac {4 \sqrt [4]{-1+x^4}}{x^4}+\sqrt {2} \arctan \left (\frac {-1+\sqrt {-1+x^4}}{\sqrt {2} \sqrt [4]{-1+x^4}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}} \left (-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 3\right ], x^{4}\right )}{8}-\left (-3 \ln \left (2\right )+\frac {\pi }{2}-1+4 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )-\frac {4 \Gamma \left (\frac {3}{4}\right )}{x^{4}}\right )}{16 \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}}}\) | \(72\) |
risch | \(-\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{4 x^{4}}+\frac {{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {3}{4}} \left (\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+4 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{16 \Gamma \left (\frac {3}{4}\right ) \operatorname {signum}\left (x^{4}-1\right )^{\frac {3}{4}}}\) | \(76\) |
pseudoelliptic | \(\frac {\ln \left (\frac {-\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}-\sqrt {x^{4}-1}-1}{\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}-\sqrt {x^{4}-1}-1}\right ) \sqrt {2}\, x^{4}+2 \arctan \left (\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}+1\right ) \sqrt {2}\, x^{4}+2 \arctan \left (\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}-1\right ) \sqrt {2}\, x^{4}-8 \left (x^{4}-1\right )^{\frac {1}{4}}}{32 x^{4}}\) | \(116\) |
trager | \(-\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{4 x^{4}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \left (x^{4}-1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}}\right )}{16}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )-2 \left (x^{4}-1\right )^{\frac {3}{4}}}{x^{4}}\right )}{16}\) | \(171\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=\frac {\left (i + 1\right ) \, \sqrt {2} x^{4} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) - \left (i - 1\right ) \, \sqrt {2} x^{4} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) + \left (i - 1\right ) \, \sqrt {2} x^{4} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) - \left (i + 1\right ) \, \sqrt {2} x^{4} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right ) - 8 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{32 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=- \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{4}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{32} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) - \frac {1}{32} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) - \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{4 \, x^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{32} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) - \frac {1}{32} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) - \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{4 \, x^{4}} \]
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Time = 5.82 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx=-\frac {{\left (x^4-1\right )}^{1/4}}{4\,x^4}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{16}-\frac {1}{16}{}\mathrm {i}\right ) \]
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