Optimal. Leaf size=93 \[ \frac {\sqrt [4]{x^2-1}}{2 x^2}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^2-1}}{\sqrt {x^2-1}-1}\right )}{4 \sqrt {2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^2-1}}{\sqrt {x^2-1}+1}\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.11, 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, 51, 63, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\sqrt [4]{x^2-1}}{2 x^2}-\frac {3 \log \left (\sqrt {x^2-1}-\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{8 \sqrt {2}}+\frac {3 \log \left (\sqrt {x^2-1}+\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{8 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{x^2-1}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt [4]{x^2-1}+1\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 204
Rule 211
Rule 266
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (-1+x^2\right )^{3/4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{3/4} x^2} \, dx,x,x^2\right )\\ &=\frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{3/4} x} \, dx,x,x^2\right )\\ &=\frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt [4]{-1+x^2}\right )\\ &=\frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^2}\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^2}\right )\\ &=\frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )}{8 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^2}\right )}{8 \sqrt {2}}\\ &=\frac {\sqrt [4]{-1+x^2}}{2 x^2}-\frac {3 \log \left (1-\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}}\\ &=\frac {\sqrt [4]{-1+x^2}}{2 x^2}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+x^2}\right )}{4 \sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} \sqrt [4]{-1+x^2}+\sqrt {-1+x^2}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 26, normalized size = 0.28 \begin {gather*} 2 \sqrt [4]{x^2-1} \, _2F_1\left (\frac {1}{4},2;\frac {5}{4};1-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 98, normalized size = 1.05 \begin {gather*} \frac {\sqrt [4]{-1+x^2}}{2 x^2}+\frac {3 \tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^2}}{\sqrt {2}}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^2}}{1+\sqrt {-1+x^2}}\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 181, normalized size = 1.95 \begin {gather*} -\frac {12 \, \sqrt {2} x^{2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1} - \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) + 12 \, \sqrt {2} x^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{2} - 1} + 4} - \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) - 3 \, \sqrt {2} x^{2} \log \left (4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{2} - 1} + 4\right ) + 3 \, \sqrt {2} x^{2} \log \left (-4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{2} - 1} + 4\right ) - 8 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{16 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 114, normalized size = 1.23 \begin {gather*} \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) + \frac {{\left (x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.61, size = 71, normalized size = 0.76
method | result | size |
meijerg | \(-\frac {\left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {3}{4}} \left (-\frac {21 \Gamma \left (\frac {3}{4}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3\right ], x^{2}\right )}{32}-\frac {3 \left (\frac {1}{3}-3 \ln \relax (2)+\frac {\pi }{2}+2 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )}{4}+\frac {\Gamma \left (\frac {3}{4}\right )}{x^{2}}\right )}{2 \Gamma \left (\frac {3}{4}\right ) \mathrm {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\) | \(71\) |
risch | \(\frac {\left (x^{2}-1\right )^{\frac {1}{4}}}{2 x^{2}}+\frac {3 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {3}{4}} \left (\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{2} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{2}\right )}{4}+\left (-3 \ln \relax (2)+\frac {\pi }{2}+2 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{8 \Gamma \left (\frac {3}{4}\right ) \mathrm {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\) | \(76\) |
trager | \(\frac {\left (x^{2}-1\right )^{\frac {1}{4}}}{2 x^{2}}+\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (x^{2}-1\right )^{\frac {3}{4}}-2 \left (x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{2}}\right )}{8}-\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {x^{2}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \left (x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{2}}\right )}{8}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 114, normalized size = 1.23 \begin {gather*} \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) - \frac {3}{16} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{2} - 1} + 1\right ) + \frac {{\left (x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 57, normalized size = 0.61 \begin {gather*} \frac {{\left (x^2-1\right )}^{1/4}}{2\,x^2}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^2-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{8}+\frac {3}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^2-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{8}-\frac {3}{8}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.95, size = 34, normalized size = 0.37 \begin {gather*} - \frac {\Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 x^{\frac {7}{2}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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