Optimal. Leaf size=54 \[ \frac {1}{12} \tan ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right )+\frac {\sqrt {x^6-1} \left (8 x^9-3 x^6-8 x^3+6\right )}{72 x^{12}} \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1475, 835, 807, 266, 47, 63, 203} \begin {gather*} \frac {\sqrt {x^6-1}}{24 x^6}-\frac {1}{24} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\left (x^6-1\right )^{3/2}}{12 x^{12}}+\frac {\left (x^6-1\right )^{3/2}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 203
Rule 266
Rule 807
Rule 835
Rule 1475
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{13}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-1+x) \sqrt {-1+x^2}}{x^5} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {(4-x) \sqrt {-1+x^2}}{x^4} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {\sqrt {-1+x^6}}{24 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{12 x^{12}}+\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 37, normalized size = 0.69 \begin {gather*} -\frac {\left (x^6-1\right )^{3/2} \left (x^9 \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};1-x^6\right )-1\right )}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 56, normalized size = 1.04 \begin {gather*} \frac {\sqrt {-1+x^6} \left (6-8 x^3-3 x^6+8 x^9\right )}{72 x^{12}}+\frac {1}{12} \tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 56, normalized size = 1.04 \begin {gather*} -\frac {6 \, x^{12} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 8 \, x^{12} - {\left (8 \, x^{9} - 3 \, x^{6} - 8 \, x^{3} + 6\right )} \sqrt {x^{6} - 1}}{72 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{x^{13}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 47, normalized size = 0.87
method | result | size |
risch | \(\frac {8 x^{15}-3 x^{12}-16 x^{9}+9 x^{6}+8 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}+\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{24}\) | \(47\) |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (8 x^{9}-3 x^{6}-8 x^{3}+6\right )}{72 x^{12}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\) | \(60\) |
meijerg | \(\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (x^{12}-8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right ) \sqrt {-x^{6}+1}}{8 x^{12}}-\frac {\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{2}+\frac {\left (\frac {1}{2}-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{x^{12}}-\frac {\sqrt {\pi }}{x^{6}}\right )}{12 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \left (-x^{6}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, x^{9}}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 58, normalized size = 1.07 \begin {gather*} -\frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}} - \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} - \frac {1}{24} \, \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 = 1.36, size = 46, normalized size = 0.85 \begin {gather*} \frac {\frac {\sqrt {x^6-1}}{24}-\frac {{\left (x^6-1\right )}^{3/2}}{24}}{x^{12}}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{24}+\frac {{\left (x^6-1\right )}^{3/2}}{9\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.82, size = 56, normalized size = 1.04 \begin {gather*} \frac {\begin {cases} \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}}{3} - \frac {\begin {cases} \frac {\operatorname {acos}{\left (\frac {1}{x^{3}} \right )}}{8} - \frac {\left (-1 + \frac {2}{x^{6}}\right ) \sqrt {1 - \frac {1}{x^{6}}}}{8 x^{3}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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