Optimal. Leaf size=35 \[ \frac{\sqrt{-x^3-1}}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
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Rubi [A] time = 0.0147958, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ \frac{\sqrt{-x^3-1}}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{-1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x^2} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{3 x^3}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{3 x^3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-1-x^3}\right )\\ &=\frac{\sqrt{-1-x^3}}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (\sqrt{-1-x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.0225215, size = 40, normalized size = 1.14 \[ \frac{1}{3} \sqrt{-x^3-1} \left (\frac{1}{x^3}-\frac{\tanh ^{-1}\left (\sqrt{x^3+1}\right )}{\sqrt{x^3+1}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 28, normalized size = 0.8 \begin{align*} -{\frac{1}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{1}{3\,{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 = 1.53544, size = 36, normalized size = 1.03 \begin{align*} \frac{\sqrt{-x^{3} - 1}}{3 \, x^{3}} - \frac{1}{3} \, \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 = 1.62839, size = 76, normalized size = 2.17 \begin{align*} -\frac{x^{3} \arctan \left (\sqrt{-x^{3} - 1}\right ) - \sqrt{-x^{3} - 1}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.45636, size = 29, normalized size = 0.83 \begin{align*} - \frac{i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{i \sqrt{1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09588, size = 36, normalized size = 1.03 \begin{align*} \frac{\sqrt{-x^{3} - 1}}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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