Optimal. Leaf size=43 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {3} (2-x) \sqrt {x}}{2 \sqrt {3 x-3 x^2+x^3}}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2022, 1927,
212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {3} (2-x) \sqrt {x}}{2 \sqrt {x^3-3 x^2+3 x}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1927
Rule 2022
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 \text {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*}
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Mathematica [A]
time = 0.08, size = 61, normalized size = 1.42 \begin {gather*} \frac {2 \sqrt {x} \sqrt {3-3 x+x^2} \tanh ^{-1}\left (\frac {x-\sqrt {3-3 x+x^2}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {x \left (3-3 x+x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 50, normalized size = 1.16
method | result | size |
default | \(\frac {\sqrt {x}\, \sqrt {x^{2}-3 x +3}\, \sqrt {3}\, \arctanh \left (\frac {\left (x -2\right ) \sqrt {3}}{2 \sqrt {x^{2}-3 x +3}}\right )}{3 \sqrt {x \left (x^{2}-3 x +3\right )}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 49, normalized size = 1.14 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.15, size = 47, normalized size = 1.09 \begin {gather*} \frac {1}{3} \, \sqrt {3} \log \left (x + \sqrt {3} - \sqrt {x^{2} - 3 \, x + 3}\right ) - \frac {1}{3} \, \sqrt {3} \log \left (-x + \sqrt {3} + \sqrt {x^{2} - 3 \, x + 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x}\,\sqrt {x\,\left (x^2-3\,x+3\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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