Optimal. Leaf size=53 \[ -\frac{\sqrt{-x^3-1}}{4 x^3}+\frac{\sqrt{-x^3-1}}{6 x^6}+\frac{1}{4} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
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Rubi [A] time = 0.0198301, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ -\frac{\sqrt{-x^3-1}}{4 x^3}+\frac{\sqrt{-x^3-1}}{6 x^6}+\frac{1}{4} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^7 \sqrt{-1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x^3} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{6 x^6}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x^2} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{6 x^6}-\frac{\sqrt{-1-x^3}}{4 x^3}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{6 x^6}-\frac{\sqrt{-1-x^3}}{4 x^3}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-1-x^3}\right )\\ &=\frac{\sqrt{-1-x^3}}{6 x^6}-\frac{\sqrt{-1-x^3}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{-1-x^3}\right )\\ \end{align*}
Mathematica [C] time = 0.0050788, size = 28, normalized size = 0.53 \[ \frac{2}{3} \sqrt{-x^3-1} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};x^3+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{4}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{1}{6\,{x}^{6}}\sqrt{-{x}^{3}-1}}-{\frac{1}{4\,{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 = 1.53207, size = 76, normalized size = 1.43 \begin{align*} -\frac{3 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{-x^{3} - 1}}{12 \,{\left (2 \, x^{3} -{\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac{1}{4} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69909, size = 95, normalized size = 1.79 \begin{align*} \frac{3 \, x^{6} \arctan \left (\sqrt{-x^{3} - 1}\right ) -{\left (3 \, x^{3} - 2\right )} \sqrt{-x^{3} - 1}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.25871, size = 66, normalized size = 1.25 \begin{align*} \frac{i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} - \frac{i}{4 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{i}{12 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{i}{6 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10089, size = 55, normalized size = 1.04 \begin{align*} \frac{3 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{-x^{3} - 1}}{12 \, x^{6}} + \frac{1}{4} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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