Optimal. Leaf size=43 \[ \frac {1}{48} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (3 x^{12}+2 x^6-8\right )}{144 x^{18}} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 47, 51, 63, 203} \begin {gather*} \frac {\sqrt {x^6-1}}{48 x^6}+\frac {1}{48} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\sqrt {x^6-1}}{18 x^{18}}+\frac {\sqrt {x^6-1}}{72 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^6}}{x^{19}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^4} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {1}{36} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {\sqrt {-1+x^6}}{48 x^6}+\frac {1}{96} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {\sqrt {-1+x^6}}{48 x^6}+\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {\sqrt {-1+x^6}}{18 x^{18}}+\frac {\sqrt {-1+x^6}}{72 x^{12}}+\frac {\sqrt {-1+x^6}}{48 x^6}+\frac {1}{48} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.65 \begin {gather*} \frac {1}{9} \left (x^6-1\right )^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};1-x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (-8+2 x^6+3 x^{12}\right )}{144 x^{18}}+\frac {1}{48} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 39, normalized size = 0.91 \begin {gather*} \frac {3 \, x^{18} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (3 \, x^{12} + 2 \, x^{6} - 8\right )} \sqrt {x^{6} - 1}}{144 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 44, normalized size = 1.02 \begin {gather*} \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{6} - 1}}{144 \, x^{18}} + \frac {1}{48} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 37, normalized size = 0.86
method | result | size |
risch | \(\frac {3 x^{18}-x^{12}-10 x^{6}+8}{144 x^{18} \sqrt {x^{6}-1}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{48}\) | \(37\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (3 x^{12}+2 x^{6}-8\right )}{144 x^{18}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{48}\) | \(55\) |
meijerg | \(\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {2 \sqrt {\pi }}{3 x^{18}}+\frac {\sqrt {\pi }}{2 x^{12}}+\frac {\sqrt {\pi }}{4 x^{6}}-\frac {\left (\frac {5}{6}-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (20 x^{18}-48 x^{12}-96 x^{6}+128\right )}{192 x^{18}}-\frac {\sqrt {\pi }\, \left (-48 x^{12}-32 x^{6}+128\right ) \sqrt {-x^{6}+1}}{192 x^{18}}+\frac {\ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }}{4}\right )}{12 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 66, normalized size = 1.53 \begin {gather*} \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{6} - 1}}{144 \, {\left (3 \, x^{6} + {\left (x^{6} - 1\right )}^{3} + 3 \, {\left (x^{6} - 1\right )}^{2} - 2\right )}} + \frac {1}{48} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 47, normalized size = 1.09 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{48}-\frac {\sqrt {x^6-1}}{48\,x^{18}}+\frac {{\left (x^6-1\right )}^{3/2}}{18\,x^{18}}+\frac {{\left (x^6-1\right )}^{5/2}}{48\,x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.33, size = 160, normalized size = 3.72 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {1}{x^{3}} \right )}}{48} - \frac {i}{48 x^{3} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {i}{144 x^{9} \sqrt {-1 + \frac {1}{x^{6}}}} + \frac {5 i}{72 x^{15} \sqrt {-1 + \frac {1}{x^{6}}}} - \frac {i}{18 x^{21} \sqrt {-1 + \frac {1}{x^{6}}}} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\operatorname {asin}{\left (\frac {1}{x^{3}} \right )}}{48} + \frac {1}{48 x^{3} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {1}{144 x^{9} \sqrt {1 - \frac {1}{x^{6}}}} - \frac {5}{72 x^{15} \sqrt {1 - \frac {1}{x^{6}}}} + \frac {1}{18 x^{21} \sqrt {1 - \frac {1}{x^{6}}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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