Optimal. Leaf size=71 \[ -\frac{5 \sqrt{1-x^3}}{24 x^3}-\frac{5 \sqrt{1-x^3}}{36 x^6}-\frac{\sqrt{1-x^3}}{9 x^9}-\frac{5}{24} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]
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Rubi [A] time = 0.0280834, antiderivative size = 71, normalized size of antiderivative = 1., 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, 206} \[ -\frac{5 \sqrt{1-x^3}}{24 x^3}-\frac{5 \sqrt{1-x^3}}{36 x^6}-\frac{\sqrt{1-x^3}}{9 x^9}-\frac{5}{24} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 206
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} \tanh ^{-1}\left (\sqrt{1-x^3}\right )\\ \end{align*}
Mathematica [C] time = 0.0048222, size = 30, normalized size = 0.42 \[ -\frac{2}{3} \sqrt{1-x^3} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};1-x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 56, normalized size = 0.8 \begin{align*} -{\frac{5}{24}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) }-{\frac{1}{9\,{x}^{9}}\sqrt{-{x}^{3}+1}}-{\frac{5}{36\,{x}^{6}}\sqrt{-{x}^{3}+1}}-{\frac{5}{24\,{x}^{3}}\sqrt{-{x}^{3}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996317, size = 122, normalized size = 1.72 \begin{align*} -\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}{48} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{5}{48} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44787, size = 161, normalized size = 2.27 \begin{align*} -\frac{15 \, x^{9} \log \left (\sqrt{-x^{3} + 1} + 1\right ) - 15 \, x^{9} \log \left (\sqrt{-x^{3} + 1} - 1\right ) + 2 \,{\left (15 \, x^{6} + 10 \, x^{3} + 8\right )} \sqrt{-x^{3} + 1}}{144 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.22443, size = 182, normalized size = 2.56 \begin{align*} \begin{cases} - \frac{5 \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} + \frac{5}{24 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{5}{72 x^{\frac{9}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{36 x^{\frac{15}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{9 x^{\frac{21}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} & \text{for}\: \frac{1}{\left |{x^{3}}\right |} > 1 \\\frac{5 i \operatorname{asin}{\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}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11308, size = 103, normalized size = 1.45 \begin{align*} -\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}{48} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{5}{48} \, \log \left ({\left | \sqrt{-x^{3} + 1} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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