Optimal. Leaf size=71 \[ \frac {5 \sqrt {-x^3-1}}{24 x^3}-\frac {5}{24} \tan ^{-1}\left (\sqrt {-x^3-1}\right )+\frac {\sqrt {-x^3-1}}{9 x^9}-\frac {5 \sqrt {-x^3-1}}{36 x^6} \]
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Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ \frac {5 \sqrt {-x^3-1}}{24 x^3}-\frac {5 \sqrt {-x^3-1}}{36 x^6}+\frac {\sqrt {-x^3-1}}{9 x^9}-\frac {5}{24} \tan ^{-1}\left (\sqrt {-x^3-1}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 204
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^{10} \sqrt {-1-x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x} x^4} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1-x^3}}{9 x^9}-\frac {5}{18} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x} x^3} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1-x^3}}{9 x^9}-\frac {5 \sqrt {-1-x^3}}{36 x^6}+\frac {5}{24} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x} x^2} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1-x^3}}{9 x^9}-\frac {5 \sqrt {-1-x^3}}{36 x^6}+\frac {5 \sqrt {-1-x^3}}{24 x^3}-\frac {5}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x} x} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1-x^3}}{9 x^9}-\frac {5 \sqrt {-1-x^3}}{36 x^6}+\frac {5 \sqrt {-1-x^3}}{24 x^3}+\frac {5}{24} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {-1-x^3}\right )\\ &=\frac {\sqrt {-1-x^3}}{9 x^9}-\frac {5 \sqrt {-1-x^3}}{36 x^6}+\frac {5 \sqrt {-1-x^3}}{24 x^3}-\frac {5}{24} \tan ^{-1}\left (\sqrt {-1-x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.39 \[ -\frac {2}{3} \sqrt {-x^3-1} \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};x^3+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 44, normalized size = 0.62 \[ -\frac {15 \, x^{9} \arctan \left (\sqrt {-x^{3} - 1}\right ) - {\left (15 \, x^{6} - 10 \, x^{3} + 8\right )} \sqrt {-x^{3} - 1}}{72 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 59, normalized size = 0.83 \[ \frac {15 \, {\left (x^{3} + 1\right )}^{2} \sqrt {-x^{3} - 1} + 40 \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-x^{3} - 1}}{72 \, x^{9}} - \frac {5}{24} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 56, normalized size = 0.79 \[ -\frac {5 \arctan \left (\sqrt {-x^{3}-1}\right )}{24}+\frac {5 \sqrt {-x^{3}-1}}{24 x^{3}}-\frac {5 \sqrt {-x^{3}-1}}{36 x^{6}}+\frac {\sqrt {-x^{3}-1}}{9 x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 74, normalized size = 1.04 \[ \frac {15 \, {\left (-x^{3} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (-x^{3} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {-x^{3} - 1}}{72 \, {\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \, {\left (x^{3} + 1\right )}^{2} + 2\right )}} - \frac {5}{24} \, \arctan \left (\sqrt {-x^{3} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 223, normalized size = 3.14 \[ \frac {5\,\sqrt {-x^3-1}}{24\,x^3}-\frac {5\,\sqrt {-x^3-1}}{36\,x^6}+\frac {\sqrt {-x^3-1}}{9\,x^9}+\frac {5\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{8\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.48, size = 90, normalized size = 1.27 \[ - \frac {5 i \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{24} + \frac {5 i}{24 x^{\frac {3}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {5 i}{72 x^{\frac {9}{2}} \sqrt {1 + \frac {1}{x^{3}}}} - \frac {i}{36 x^{\frac {15}{2}} \sqrt {1 + \frac {1}{x^{3}}}} + \frac {i}{9 x^{\frac {21}{2}} \sqrt {1 + \frac {1}{x^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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