Optimal. Leaf size=43 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (2-x) \sqrt{x}}{2 \sqrt{x^3-3 x^2+3 x}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0474514, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1997, 1913, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (2-x) \sqrt{x}}{2 \sqrt{x^3-3 x^2+3 x}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1997
Rule 1913
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \sqrt{x \left (3-3 x+x^2\right )}} \, dx &=\int \frac{1}{\sqrt{x} \sqrt{3 x-3 x^2+x^3}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{(6-3 x) \sqrt{x}}{\sqrt{3 x-3 x^2+x^3}}\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} (2-x) \sqrt{x}}{2 \sqrt{3 x-3 x^2+x^3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0197291, size = 62, normalized size = 1.44 \[ \frac{\sqrt{x} \sqrt{x^2-3 x+3} \tanh ^{-1}\left (\frac{\sqrt{3} (x-2)}{2 \sqrt{x^2-3 x+3}}\right )}{\sqrt{3} \sqrt{x \left (x^2-3 x+3\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 50, normalized size = 1.2 \begin{align*}{\frac{\sqrt{3}}{3}\sqrt{x}\sqrt{{x}^{2}-3\,x+3}{\it Artanh} \left ({\frac{ \left ( -2+x \right ) \sqrt{3}}{2}{\frac{1}{\sqrt{{x}^{2}-3\,x+3}}}} \right ){\frac{1}{\sqrt{x \left ({x}^{2}-3\,x+3 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (x^{2} - 3 \, x + 3\right )} x} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39325, size = 132, normalized size = 3.07 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{7 \, x^{3} + 4 \, \sqrt{3} \sqrt{x^{3} - 3 \, x^{2} + 3 \, x}{\left (x - 2\right )} \sqrt{x} - 24 \, x^{2} + 24 \, x}{x^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14413, size = 69, normalized size = 1.6 \begin{align*} -\frac{1}{3} \, \sqrt{3} \log \left ({\left | -x + \sqrt{3} + \sqrt{x^{2} - 3 \, x + 3} \right |}\right ) + \frac{1}{3} \, \sqrt{3} \log \left ({\left | -x - \sqrt{3} + \sqrt{x^{2} - 3 \, x + 3} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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