Optimal. Leaf size=90 \[ \frac {2}{3} \sqrt [4]{x^6-1}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^6-1}}{\sqrt {x^6-1}-1}\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^6-1}}{\sqrt {x^6-1}+1}\right )}{3 \sqrt {2}} \]
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Rubi [A] time = 0.11, antiderivative size = 142, normalized size of antiderivative = 1.58, 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, 50, 63, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {2}{3} \sqrt [4]{x^6-1}+\frac {\log \left (\sqrt {x^6-1}-\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{6 \sqrt {2}}-\frac {\log \left (\sqrt {x^6-1}+\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{6 \sqrt {2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )}{3 \sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )}{3 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
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^6}}{x} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt [4]{-1+x}}{x} \, dx,x,x^6\right )\\ &=\frac {2}{3} \sqrt [4]{-1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{3/4} x} \, dx,x,x^6\right )\\ &=\frac {2}{3} \sqrt [4]{-1+x^6}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right )\\ &=\frac {2}{3} \sqrt [4]{-1+x^6}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^6}\right )\\ &=\frac {2}{3} \sqrt [4]{-1+x^6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )}{6 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^6}\right )}{6 \sqrt {2}}\\ &=\frac {2}{3} \sqrt [4]{-1+x^6}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}\\ &=\frac {2}{3} \sqrt [4]{-1+x^6}+\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+x^6}\right )}{3 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^6}+\sqrt {-1+x^6}\right )}{6 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 135, normalized size = 1.50 \begin {gather*} \frac {1}{12} \left (8 \sqrt [4]{x^6-1}+\sqrt {2} \log \left (\sqrt {x^6-1}-\sqrt {2} \sqrt [4]{x^6-1}+1\right )-\sqrt {2} \log \left (\sqrt {x^6-1}+\sqrt {2} \sqrt [4]{x^6-1}+1\right )+2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{x^6-1}\right )-2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt [4]{x^6-1}+1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 95, normalized size = 1.06 \begin {gather*} \frac {2}{3} \sqrt [4]{-1+x^6}-\frac {\tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^6}}{\sqrt {2}}}{\sqrt [4]{-1+x^6}}\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^6}}{1+\sqrt {-1+x^6}}\right )}{3 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 164, normalized size = 1.82 \begin {gather*} \frac {1}{3} \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1} - \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} - 1\right ) + \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} - 1} + 4} - \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{12} \, \sqrt {2} \log \left (4 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} - 1} + 4\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-4 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{6} - 1} + 4\right ) + \frac {2}{3} \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 111, normalized size = 1.23 \begin {gather*} -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 11.70, size = 64, normalized size = 0.71
method | result | size |
meijerg | \(-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{4}} \left (\Gamma \left (\frac {3}{4}\right ) x^{6} \hypergeom \left (\left [\frac {3}{4}, 1, 1\right ], \left [2, 2\right ], x^{6}\right )-4 \left (4-3 \ln \relax (2)+\frac {\pi }{2}+6 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )\right )}{24 \Gamma \left (\frac {3}{4}\right ) \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{4}}}\) | \(64\) |
trager | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{3}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+2 \left (x^{6}-1\right )^{\frac {3}{4}}+2 \sqrt {x^{6}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )+2 \left (x^{6}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{6}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}-2 \sqrt {x^{6}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (x^{6}-1\right )^{\frac {3}{4}}-2 \left (x^{6}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{6}}\right )}{6}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 111, normalized size = 1.23 \begin {gather*} -\frac {1}{6} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{6} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{6} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{6} - 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 54, normalized size = 0.60 \begin {gather*} \frac {2\,{\left (x^6-1\right )}^{1/4}}{3}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^6-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.86, size = 39, normalized size = 0.43 \begin {gather*} - \frac {x^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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