Optimal. Leaf size=31 \[ \frac{\sqrt{x^3-1}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left (\sqrt{x^3-1}\right ) \]
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Rubi [A] time = 0.0107069, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 203} \[ \frac{\sqrt{x^3-1}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left (\sqrt{x^3-1}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 203
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 = 0.0156903, size = 41, normalized size = 1.32 \[ \frac{1}{3} \sqrt{x^3-1} \left (\frac{1}{x^3}+\frac{\tanh ^{-1}\left (\sqrt{1-x^3}\right )}{\sqrt{1-x^3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 24, 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 = 1.51794, size = 31, normalized size = 1. \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 = 1.71203, size = 72, normalized size = 2.32 \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 [A] time = 2.17334, size = 82, normalized size = 2.65 \begin{align*} \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{i \sqrt{-1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} & \text{for}\: \frac{1}{\left |{x^{3}}\right |} > 1 \\- \frac{\operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{1}{3 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{3 x^{\frac{9}{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 = 1.17929, size = 31, normalized size = 1. \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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