Optimal. Leaf size=16 \[ \frac{2}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
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Rubi [A] time = 0.0083577, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 204} \[ \frac{2}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{-1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x} \, dx,x,x^3\right )\\ &=-\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-1-x^3}\right )\right )\\ &=\frac{2}{3} \tan ^{-1}\left (\sqrt{-1-x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.0023372, size = 16, normalized size = 1. \[ \frac{2}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 13, normalized size = 0.8 \begin{align*}{\frac{2}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50993, size = 16, normalized size = 1. \begin{align*} \frac{2}{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 = 1.70377, size = 38, normalized size = 2.38 \begin{align*} \frac{2}{3} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.97644, size = 12, normalized size = 0.75 \begin{align*} \frac{2 i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15346, size = 16, normalized size = 1. \begin{align*} \frac{2}{3} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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