Optimal. Leaf size=49 \[ \frac {\left (-2 x^6-3 x^3+2\right ) \sqrt {x^6-1}}{18 x^9}-\frac {1}{3} \tan ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 47, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1475, 807, 266, 47, 63, 203} \begin {gather*} -\frac {\sqrt {x^6-1}}{6 x^6}+\frac {1}{6} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\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 1475
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^{10}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-1+x) \sqrt {-1+x^2}}{x^4} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=-\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {\left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{6} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 1.33 \begin {gather*} -\frac {x^6+\sqrt {1-x^6} x^6 \tanh ^{-1}\left (\sqrt {1-x^6}\right )-1}{6 x^6 \sqrt {x^6-1}}-\frac {\left (x^6-1\right )^{3/2}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 51, normalized size = 1.04 \begin {gather*} \frac {\left (-2 x^6-3 x^3+2\right ) \sqrt {x^6-1}}{18 x^9}-\frac {1}{3} \tan ^{-1}\left (\frac {\sqrt {x^6-1}}{x^3-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 51, normalized size = 1.04 \begin {gather*} \frac {6 \, x^{9} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 2 \, x^{9} - {\left (2 \, x^{6} + 3 \, x^{3} - 2\right )} \sqrt {x^{6} - 1}}{18 \, x^{9}} \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^{10}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 0.86 \begin {gather*} -\frac {2 x^{12}+3 x^{9}-4 x^{6}-3 x^{3}+2}{18 x^{9} \sqrt {x^{6}-1}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 35, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} + \frac {1}{6} \, \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 = 1.10, size = 35, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{6}-\frac {\sqrt {x^6-1}}{6\,x^6}-\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 = 4.25, size = 49, normalized size = 1.00 \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 )}}{2} - \frac {\sqrt {1 - \frac {1}{x^{6}}}}{2 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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