Optimal. Leaf size=28 \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
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Rubi [A] time = 0.0094395, antiderivative size = 28, 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, 50, 63, 207} \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x^3}}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x} \, dx,x,x^3\right )\\ &=\frac{2 \sqrt{1+x^3}}{3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^3\right )\\ &=\frac{2 \sqrt{1+x^3}}{3}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^3}\right )\\ &=\frac{2 \sqrt{1+x^3}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{1+x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.0054956, size = 28, normalized size = 1. \[ \frac{2 \sqrt{x^3+1}}{3}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 21, normalized size = 0.8 \begin{align*} -{\frac{2}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }+{\frac{2}{3}\sqrt{{x}^{3}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953164, size = 46, normalized size = 1.64 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} - \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 [A] time = 1.88637, size = 104, normalized size = 3.71 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} - \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 = 1.12455, size = 48, normalized size = 1.71 \begin{align*} \frac{2 x^{\frac{3}{2}}}{3 \sqrt{1 + \frac{1}{x^{3}}}} - \frac{2 \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{2}{3 x^{\frac{3}{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 = 1.09201, size = 47, normalized size = 1.68 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} - \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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