Optimal. Leaf size=93 \[ -\frac {\sqrt [4]{x^4-1}}{4 x^4}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^4-1}}{\sqrt {x^4-1}-1}\right )}{8 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^4-1}}{\sqrt {x^4-1}+1}\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.12, antiderivative size = 145, normalized size of antiderivative = 1.56, number of steps used = 12, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {266, 47, 63, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\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}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{8 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 204
Rule 211
Rule 266
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-1+x^4}}{x^5} \, dx &=\frac {1}{4} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{8} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )-\frac {\operatorname {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 {\operatorname {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 {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{8 \sqrt {2}}-\frac {\operatorname {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 {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{8 \sqrt {2}}+\frac {\tan ^{-1}\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*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.30 \begin {gather*} \frac {1}{5} \left (x^4-1\right )^{5/4} \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};1-x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 98, normalized size = 1.05 \begin {gather*} -\frac {\sqrt [4]{-1+x^4}}{4 x^4}+\frac {\tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^4}}{\sqrt {2}}}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 180, normalized size = 1.94 \begin {gather*} -\frac {4 \, \sqrt {2} x^{4} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1} - \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} - 1\right ) + 4 \, \sqrt {2} x^{4} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1} + 4} - \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 1\right ) - \sqrt {2} x^{4} \log \left (4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1} + 4\right ) + \sqrt {2} x^{4} \log \left (-4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1} + 4\right ) + 8 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{32 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 114, normalized size = 1.23 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 7.89, size = 72, normalized size = 0.77
method | result | size |
meijerg | \(\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} \left (-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 3\right ], x^{4}\right )}{8}-\left (-3 \ln \relax (2)+\frac {\pi }{2}-1+4 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )-\frac {4 \Gamma \left (\frac {3}{4}\right )}{x^{4}}\right )}{16 \Gamma \left (\frac {3}{4}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}}}\) | \(72\) |
risch | \(-\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{4 x^{4}}+\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {3}{4}} \left (\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{4} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4}+\left (-3 \ln \relax (2)+\frac {\pi }{2}+4 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{16 \Gamma \left (\frac {3}{4}\right ) \mathrm {signum}\left (x^{4}-1\right )^{\frac {3}{4}}}\) | \(76\) |
trager | \(-\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{4 x^{4}}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \left (x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}}\right )}{16}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 114, normalized size = 1.23 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 57, normalized size = 0.61 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.98, size = 34, normalized size = 0.37 \begin {gather*} - \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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