Optimal. Leaf size=16 \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]
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Rubi [A] time = 0.0082425, 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, 206} \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 206
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} \tanh ^{-1}\left (\sqrt{1-x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.0022916, size = 16, normalized size = 1. \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 13, normalized size = 0.8 \begin{align*} -{\frac{2}{3}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.972795, size = 39, normalized size = 2.44 \begin{align*} -\frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{3} \, \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 [B] time = 2.15565, size = 81, normalized size = 5.06 \begin{align*} -\frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.99846, size = 31, normalized size = 1.94 \begin{align*} \begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \frac{1}{\left |{x^{3}}\right |} > 1 \\\frac{2 i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07393, size = 41, normalized size = 2.56 \begin{align*} -\frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{3} \, \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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