Optimal. Leaf size=39 \[ -\frac {1}{4} \sqrt {x^2-x} (1-2 x)-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {612, 620, 206} \[ -\frac {1}{4} \sqrt {x^2-x} (1-2 x)-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rubi steps
\begin {align*} \int \sqrt {-x+x^2} \, dx &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{8} \int \frac {1}{\sqrt {-x+x^2}} \, dx\\ &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )\\ &=-\frac {1}{4} (1-2 x) \sqrt {-x+x^2}-\frac {1}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.18 \[ \frac {2 x^3-3 x^2+x+\sqrt {-((x-1) x)} \sin ^{-1}\left (\sqrt {1-x}\right )}{4 \sqrt {(x-1) x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.87, size = 36, normalized size = 0.92 \[ \frac {1}{4} \, \sqrt {x^{2} - x} {\left (2 \, x - 1\right )} + \frac {1}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 37, normalized size = 0.95 \[ \frac {1}{4} \, \sqrt {x^{2} - x} {\left (2 \, x - 1\right )} + \frac {1}{8} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 0.85 \[ -\frac {\ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x}\right )}{8}+\frac {\left (2 x -1\right ) \sqrt {x^{2}-x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 43, normalized size = 1.10 \[ \frac {1}{2} \, \sqrt {x^{2} - x} x - \frac {1}{4} \, \sqrt {x^{2} - x} - \frac {1}{8} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 29, normalized size = 0.74 \[ \sqrt {x^2-x}\,\left (\frac {x}{2}-\frac {1}{4}\right )-\frac {\ln \left (x+\sqrt {x\,\left (x-1\right )}-\frac {1}{2}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{2} - x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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